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LOGARITHMIC* ARITHMETIC*
CONTAINING A NEW .AU^OQRRECT
TABLE OF Lof^THMS
OF THE NATURAL NUMBERS FROM 1 TO 10.000,
EXTENDED TO SEVEN PLACES BESIDES THE INDEX J AND SO
CONTRIVED, THAT THE LOGARITHM MAY BE EASILY
FOUND TO ANY NUMBER BETWEEN 1 AND
10,000,000.
ALSO,
AN EASY METHOD
OF CONSTRUCTING A TABLE OF LOGARITHMS, TOGETHER
WITH THEIR NUMEROUS AND IMPORTANT USES
IN THE MORE DIFFICULT PARTS OF
ARITHMETICK.
TO WHICH ARE ADDED A NUMBER OF
ASTRONOMICAL TABLES,
BY WHICH THE DIFFERENT PHASES OF THE MOON,— THE
TIMES OF HER OPPOSITION AND CONJUNCTION, MAY
BE COMPUTED WITH THE OREATEST EASE AND
exactness: and an easy method
of calculating £.
SOLAR AND LUNAR ECLIPSES j *
V "
ILLUSTRATED WITH
.*•■•..
GEOMETRICAL PROJECTIONS:
DESIGNED FOR THE INSTRUCTION OF YOUTH IN THE
SCHOOLS AND ACADEMIES
OF NEW ENGLAND,
BY ELIJAH HINSDALE JURRITT.
WILL I A MSB VR Iff} ;
PRINTED BY KPKRAIM WHITMAN.
161&
"7*)
DISTRICT OF MASSACHUSETTS, to wit *
District Clerk* s*Qj/itt.
Be it remembered, That on the fifteenth day of
September, A. D. 1818, and in the forty-third year of the Inde*
pendence of the United States of America, Elijah Hinsdale
Burritt, of the said district, has deposited in this office the
title of a book, the right whereof he claims as author, in the
words following, to wit ;
Logarithmick Arithmetick, containing a new and correct Ta*
ble of Logarithms of the natural numbers from 1 to 10,000, ex*
tended to seven places besides the index ; and so contrived, that
the Logarithm may be easily found to any number between I
and 10,000,000. Also, an easy method of constructing a Table
of Logarithms, together with their numerous and important
uses in the more difficult parts of Arithmetick. To which are
added a number of Astronomic! Tables, by which' the differ-
ent phases of the Moon, the times of her opposition and con-
junction, may be computed with the greatest ease and exact-
ness : and an easy method of calculating' Sqlar aud Lunar
Eclipses ; illustrated with Geometrical Projections : Designed
for the instruction of Youth in the Schools and Academes of
New England. By Elijah Hinsdale Burrjtt.
In conformity to the act of the congress of the United States,
entitled, " An act for the encouragemeut of learning, by secur-
ing the copies of mapsj charts and books, to the authors and
proprietors of such copies, during the times therein mentioned s"
and ahso to an act entitled, « An act supplementary tp ajiact, en-
titled, an act for the encouragement of learning, by securing
the copies of maps, charts and hooks, to the authors and propri-
etors of such copies during the timrs therein mentioned ; and
^xtentling the benefits thereof to the arts of designing, engraving
and etching historical, and other prints." JOHN W. DAVIS,
Clerk qf the District qf Massachusetts,
J
, 1
dak
FREFA
« TlltStask cfau author ;* says Dr. Johnson, is to teach whas»
Is not known, or to recommend known truths by his manner of
adorning them." To attempt the former is sufficiently irksome
to enervate endeavour, and to succeed in the latter, he nvust sur-
mount obstcles that no sagacity can avoid, and. encounter diffi-
culties which prevision cannot alleviate. He must appeal t*
judges prepossessed by passions or biased by prejudices : to
some who tire more willing to go wrong by their own judgment,
than to be indebted lor a better or a safer way to the wisdom of
another; and toothers, whose good taste and perspicacity will
not allow them to read any thing until its reputation is ectablish-
ed by the " Ministers of Criticism." But of those who are
jnor*c inclined to be pleased* who may be persuaded to remit
their censure at those errdurs into which tho. author has inad-
vertently or unavoidably fallen, and who will pass with a friend-
ly eye over those imperfections that are inseperably connected
"with all human productions ; it is hoped* though the reader
should not find his feelings ifhperceptably stolen from him by
the enchantment of novelty, or his understanding captivated by
the profundity of invention, that if his patience can endure a care-
ful perusal of the following work, be will dismiss it with, at least
this «' negative encomium," that it is a laudable attempt to im-
prove the instruction of the unlearned, and furnish schools in
genera! with a useful and comprehensive system of Practical
Arithmetick.
He who is resolved to attain any degree of eminence in thje
science of Mathematicks, dooms himself to wade through a toil*
* some* course of severe, uriinte re sling labour, unaccompanied
with any of those charms that can engage the feelings by their
novelty, or delight the imagination by their lusture ; without
any hope of honorable reward to stimulate to exertion, or sof-
ten the asperities of study. But the writer of Common Arith-
metick has a much easier taak* The labours of many that have
" gone before him as " pioneers,*' who wore distinguised by pro-
found investigation and mathematical researches, have render-
ed his path comparatively easy, even where their feet have never
trodden ; " as the sun dissipated the shades of darkness, and
spreads a twilight beyond the immediate influence of his direct
beams." Indeed, it may be said that the only necessary rcsourse
of the writers of the present age> is to copy the best examples
of their "predecessors with such modifications and amendments
I
PREFACE.
as the different modes of reckoning in business, the fluctuation
of coins, weights, and measures, and as the invention of means
to facilitate the samfltfypgpns require.
The practical sysf^Hpr Vulgar Arithmetick already extant
are very numerous ; ana the authors of some of them have ex-
hibited great knowledge and ability, it may therefore be ex-
pected, that he who ventures to add to their number should pro-
duce a substantial plea for such an obtrusion on the publick no-
tice. It may justly be expected that he should be either an ac-
curate schollar or a profound mathematician. The writer has
no claim either to the one or the other of these distinctions ;
' and must therefore plead another apology.
Having been repeatedly solicited by a number of gentlemen
respectable for their understanding and skill in Mathematical
Science, to publish his Table of Logarithms constructed only
for private use, together with their numereus and important
uses in the more difficult parts of Arithmetick Trigonometry
&c. and believing', notwithstanding the endless variety of books
now extant on Vulvar Arithmetick, there is yet room to intro-
duce usefulness with novelty, the author of the following work
humbly ventures to*assume the responsibility of endeavouring to
avoid the redundencies of some, and to supply the deficiencies
of others.
It is believed that a more accurate and extensive System of
Logarithms for natural numbers was never before published in
this country* The best American Tables are carried only to
six places besides the index, which, though capable of giving an-
swers sufficiently exact in most solutions, are, notwithstanding,
deficient where great accuracy is required. Iff constructing
the following table the author has pursued the" Differential
Method** hinted at by Mr Henry Briggs, Professor of Geometry
in Gresham College London, and Dr. Hutton's " Practical Rule
for the Construction of Logarithms."
He was also solicited to publich with this, a Table of Loga-
rithmick Signs, Tangents, &c. carried to seven places decimal,
which should correspond with those for natural numbers; but
as thb would render the work too voluminous for a common
school-book, it was thought unadvisable to augment the expense
-without proportionably increaseing the advantage. Should their
occur sufficient reason for publishing such a system of Logar-
ithms and Logarithmick Sines and Tangents, together with oth-
er Logarithmick and Mathematical Tables, they may be given
to the publick at some future period.
The method op commuting jby Logarithms, whepe it can
be adopted, aa in the evolving of roots, is manifestly the most
expiditious of any that human powers have hitherto invented.
*TV learner, who before was unacquainted with their propevr*-
PREFACE.
ties, Who has had the patience and assiduity to toil through the
tedious course of evolution by the usual process, will admire to
find that so much labour should ever ^Mpreduced to so much
ease ! ■ - .^F
The author has endeavoured, though in some instances at the
expense of deviating from Ancient usage, to arrange the sever-
al parts of Vulgar Arithmetick according to their relative im-
portance, and their mutual dependence upon each other ; and
to render the whole as easy and familiiar as the nature of the
subjects would admit.
From what experience the author has had in the instruction
of youth, aud the general complaint of the want of some work
adapted to accompany those elementary treatises on Astronomy
which are used in our Schools and Academies, the writer was in-
duced to subjoin a short* though imperfect sketch of Practical
Astronomy; And in this, he has been more desirous to be use-
ful than to appear oiiginal. AH the Tables except the II, XVIII
and XIX, together with the method of calculating the time of
New and Full Moons and Eclipses, were taken irom Dr. Brew-
ster's edition of Fugeson's Astronomy lately published. The
method of Projecting Eclipses is purely mathematics, and is
not subject to those inacuracies which the use of the sector is
liable to produce.
Whatever merit justice may award to the following primetial
pages, the author cannot forbear repeating, that it was the hope
only of being useful, that induced their submission to publick
scrutiny. If he has failed in the attempt, the disquietude of
disappointment will -be allayed by the reflection, that he is not
the first " gut magma ecocide t ausia" who have misjudged in
their abilities. To suppoae the work wholly exempt from *r-
rours> whether of the press or of -the pen, would be to suppose
what the most vigilant care has seldom performed. And wheth-
er these will be found to have a counterbalance of good, is re-
fered to the candour of those who may read to decide ; — they
are the constituted judges, and to them he submits with defter-
ence and respect.
R H. B.
Williams College, October, 181*.
NOTATION ..--.*
Simple Addition - * - -•
, .Federal Money # - . *
Simple Subtraction - - - -
Subtraction of Federal Money- • «. .#
Simple Multiplication * - . „ * , *
Multiplication of Federal Money - -
Simple Division - * * -.
Contractions in Division - - - —
Short Division - - - - -
.Supplement to Multiplication - - -
Practical Questions in Multiplication and division
Compound Addition - . - •• *
Compound Subtraction - - - -
Compound Multiplication - - •■ • -
-i Of Weight) MeasuiCj Sterling Money Sec.
Compound Division - - - ' *
— » — Of Sterling Money, Weights, Measures, &c.
Redaction - . - -
Reduction of Currencies
Table of Coins current in the United States
Duodecimals* or Cross Multiplication
Simple Proportion or Rule of Three *
Of Logarithms -- -
Construction of Logarithms *
Another Method of Computing Logarithms
Directions for taking Logarithms arid their
numbers from the Table ' -
Method of Calculating by Logarithms - *
Multiplication by Logarithms * * -
Division by Logarithm* -' - - •■ -
Proportion by Logarithm* > • - - - ' - • *
Arithmetical Complement * - *
Vulgar Fractions - «• -> - «*
Decimal Fractions „---•*
Addition of Decimals
Subtraction of Decimals -
Multiplication of Decimals - -
' Diyision of Decimals -
'Reduction of Decimals -
Simple Interest - - -
Insureance, Commission, and Brokerage
Discount - - - - - T - -
$quation of Payments ------
IPeS'lowship -
Compound Fellowship - - -
Alligation - -
;Tare and Trett - -
Compound Proportion) or Doupje Rule of Three
Do. by Logarithms - - -
Pag&+
9
* lO
li
IS
13
14
15
16
17
- YS
- 20
SO
- 21
24
26
27
26
39
30
33
34
35-
38
44
47
49
50-
5?
52
53
54
55
36
58
59
60
61 •
62
63
65
69
99
70
71
72
73
71
77
79
CONTENTS $
Compound Interest - ~ - 80
Involution - - - - - - - 82
Evolution, or Extraction of Roots ' -,,' ' -' - - 83
Do. by Logarithms ------ 8f.
Practical Questions in Evolution and Involution - 94
Annuities, or Pensions - 100
Vulgar Fractions - - - IQ5
Reduction of Vulgar Fractions, - - - .- 106
Addition of Vulgar Fractions - . - - MO
Multiplication of Vulgar Fractions - - - lit
Division of Vulgar Fractions - - - - -Hi
Simple Proportion in Vulgar Fractions - - 112
Reduction of Decimal Fractions - - - - 1M
Simple Proportion by Decimals . - . - -. - 115
Simple Interest by Decimals - - ■» - 1 1 &
Tables shewing the Amount, and the Rebate of one Dollar,
at 6 per cent for Years and MouUra - - * 119
Construction of said Tables • -* - -■ - 120
Compound Interest by decimals - - - 122
Logarithmick Tables for Years - - - - 123
Do'. do. for Months and Da> s - - 124
Arithmetical Progression - - - 125
Creometrica I Progression - ... - 12T
Position - - ■ - - - - -• - 133
Double Position 134
Permutation of Quantities - - - 1 23
Construction of Tables belonging to Compound Interest 130
Tables relating to Compound Interest . - - 139.
Practical Astronomy - - - - « -. 4 £3
Tables for Calculating the True Time of New and Full
Moon's an4 Eclipses * 149—165
Precepts relating to the preceding Tables - - 165
Do. for Calculating the Uptime of New or Full Moon 165
To Calculate the true place of the Sun for any given Moment
of Time - - - - - 186
To know whether there is an Eclipse at the time of any New
or Full Moon - - - - -.18*
To Project an Eclipse of the Sun - <■ * 19^
To Project an Eclipse of the Sun Geometrically - \9%
Geometrical Projection of Eclipses - - - - 19$
To preject an Eclipse of the Moon - - . 196
To project an Eclipse of the Mooyi Geqmetrically * 200
To find the number of Eclipses, there are in any given Year,
and in what Months they happen - - SQ3
To find on what part of the Globe any given Eclipse of the
Sun or Moon is visible - ' - - - 204
Table of Logarithms for Natural numbers - 207 — 24f
Appendix to Logarithms - - * - *XVi— ^A
Explanation of Characters used in this book.
«= Equal to, as J 2d. =* Is. signifies that 12 pence are equal to
1 shilling.
+ More, the sign of addition, as 5 -f 7=12, signifies that 5
and 7 added together, are equal to 12.
— - Minus, or less, the sign of subtraction, as 6 • — 2=4, signi-
fies that 2 subtracted from 6, leaves 4.
X Multiply, or with the signof Multiplication ; as 4 X 3=12
signifies that 4 multiplied by 3, is equal to 12.
-r- The sign of division ; as 8 -5- 2 « 4, signifies that 8 divided
by 2, is equal to 4 ; or thus, i=4, each of which signifies
the same thing. -
:: Four points set in the middle of four numbers, denote them
to be proportional to one another, by the rule of three ; as
2 : 4 :: 8 : Id ; that is, as 2 is to 4, so is 8 to 16.
4/ Prefixed to any number, supposes that the squre root of that
number is required.
3
tf Prefixed to any number, supposes the cube root of that num-
ber is required.
4 .
4/ Denotes the biquadrate root, or fourth power, Sec.
MULTIPLICATION TABLE.
1 | 2 | 3 | 4
5 | 6 | 7 | 8 |
•1
10 |
" 1
12
2 | 4 | 6 | 8 1 10 j!2 [14 |l6 |
18 1
20
22
24
3 | 6 • 9 |12 |i5 |*8 |2i |24
a7 |
30
33
36
4 | 8 |I2 |i6 1-0 |24 |28 |32
36
40
44
48
5 jlO 15 (20
25 |30 .
36 (40
, 4S
.50
1 »
60
6 |»2
18
^4
3u |36 |42 |48
54
60
| 66
72
7
14
it
21
24
28
32
35
lo"
42 149 |56 1
48~|56~l4i4"|
63 I
70
77
I 84
8
72 |
80
88
96
9
18
27
So"
36
40
45
50
54
60
63 1/2
8.
90
9V
108
10 1
20
70 |80
90
100
L.10
120
1 1
22
33
3(T
44
Is"
55
60~
66
72
77
84"
88
99
UO
121
132
|,2
24
J6
108
120
132
144
'To learn this table : Find your multiplier iri the left hand col-
umn* and the multiplicand a-top, and in the common angle of
meeting, or against your multiplier, along at the right band, and
under your multiplicand, you will find the product, or answer..
WABLE£ OF WEIGHTS AND MEASURES,
1. Sterling Money.
4 farthing* make 1 pevny, tf.
11 pence, 1 shilling, *.
20 shillings, 1 pound, £
2. Troy Weight.
34 grains (gr.) make 1 penny-weight, marked fivft.
20 penny-weights, 1 ounce, zu.
12 ounces, 1 pound,, id.
3. Avoir dufiois Weight.
16 drams (dr.) make 1 ounce, wz.
16 ounces, 1 pound, lb,
28 pounds, I quarter of a hundred weight, ?r.
4 quarters, 1 hundred weight, owf.
20 hundred Weight, 1 ton, T.
By this weight are weighed all coarse and drossy goods, grc
eery wares, and all metals except gold and silver.
4. Cloth Measure.
4 nails fna.J make 1 quarter of a yard, qr.
4 quarters, 1 yard, yd.
3 quartters, • 1 EH Flemish, E. >V.
s quarevs, 1 Efl English, E. SB.
6 quarters, 1 Ell French, A Fr.
5. Dry Measure.
2 pints (fit.) make 1 quart, qt.
5 quarts, 1 peck, fik.
4 pecks, l bushel, bu.
This measure is applied to grain, beans, flak-seed, salt, roof
oysters, coal, &c,
6. Wine Measure.
4 gills (gL) make 1 pint, fit.
3 pints, 1 quart, qt.
4 quarts, 1 gallon, gal,
311 gallons, 1 barrel, bl.
42 gallons, 1 tierce, Her.
43 gallons, 1 hogshead hhd.
2 hogsheads, 1 pipe, fi.
3 pipes, l tun, T.
Al* brandies, spirits, mead, vinegar, oil, Sec. are measured fr
wine measure. Note. — 231 solid inches, make a gallon.
7. Long Measure.
3 barley corns (b. c.) make 1 inch, marked fa,
12 inches, 1 foot, ft.
3 feet, 1 yard, yc \.
Si yards, 1 rod, pole, or perch, rd.
40 rods, 1 furlong, fur.
8 furlongs, 1 mile, m.
3 miles, 1 leagye, i ca .
49$ statute miles, 1 degree on the earth.
340 degrees, the circumference of the cm'A*
8 TABLES OF WEIGHTS AND MEASURE*.
The i|se of kw^ measure is to measure the distance of places,
or any other thing, w^here length is considered, without regard
to breadth.
N. B. In measuring the height of horses, 4 inches make t
hand. In measuring depths, 6 feet make 1 fathom, or French
toise. Distances are measured by a chata, four rods long, con-
taining one hundred links.
8. Land or Square Measure.
144 square inches make 1 square foot.
9 square feet, 1 square yard,
30$ square yards, or ) ,
272^ square feet, J * squ * r9 10fl '
"40 square rods> 1 square rood.
i 4 square roods, 1 square acre
| 640 square acres, I square mile.
i 9 S9iid or Cubic Measures
\ 1728 solid inches make 1 solid foot.
f « r ee ! °l u 0Und • ' im u "' ° r I » ton or load.
50 feet of hewn timber, y
\ 128 solid feet, or 8 feet long, > • v cord of wood;
•«. 4 wide, and 4 high, 3
1 All solids, or things that have length, breadth and depth, are
I measured by this measure. N. B- The wine gallon contains
J 231 solid or cubic inches, and the beer gallon, 232, A1)Ushet
i contains 2150,42 solid inches. \>
f 10. Time.
60 seconds (S. ) make 1 minute, marked 8. Mi.
*0 minutes, 1 hour, h >
54 hours, I day, <*•
7 days, 1 week, w.
4 weeks, 1 month, mo.
}3 months, 1 day, and 6 hours, 1 Julian year yr.
Thirty days hath September, April, June, and November,
I February twenty-eight alone, all the rest have thirty-one.
{ N. B. In bissextile or leap-year, Febmary hath 19 days.
J 1 1 Circular motion.
\ 60 second's (") make 1 minute, marked r
: «0 minutes, 1 degree, #
! 30 degrees 1 sign, *•
{ '\% sign?, or 560 degrees, the wh*lc great circle of the Zodiac.
ARITHMETICS
»o«<
i
S THE Science of Numbers, and exhibits the method, or
art of computing by them : it is divided into five parts, viz.
Rotation, Addition, Subtraction, Multiplication, and Division.
NOTATION.
Notation teaches how to read and write numbers represent*
ed by the following characters, 1, 2, 3, 4, 5, 6, 7, 8, 9, O ; each
of which has a simple value, and also a local value, according to
the order pf their combination, as in the following table.
I
10 VULGAR
To know the value of any number tfjiguret.
RULE.
Numerate from the right hand figure to the left as in the ta-
ble, and to the simple value of each figure apply its local value,
according as it is removed from the place of units towards the
right. *
EXAMPLES.
Mead the folio wing numbers.
75 Seventy five.
937 Nine hundred and thirty seven.
46 1 7 Four thousand six hundred and seventeen.
59000028 Fifty nine millions and twenty eight.
679244321 Six hundred and seventy nine millions two hun-
dred and forty four thousand three hundred and twenty one.
To write numbers.
. m RULE.
Beginning on the right hand, write unites in their place,
tens in the tens place, and so on towards the left hand, writing
each figure according to its value in numeration, and supplying
those places of the natural order with cyphers which are omitted
in the question.
, EXAMPLES.
Write down the following numbers.
Twenty four 24
V Three hundred and five 505
Seven thousand and ninety 7090
Eight millions and eight 8000008
SIMPLE ADDITION.
Is the uniting into one number several smaller ones tf the
same denomination.
RULE.
Write the numbers distinctly, units under units, tens under
tens, &c. Then reckon the amount of the right hand column.
If it be less than ten, write it down. If it exceed ten, write the
units only and carry the tens to the next place.
In like manner carry the tens of each column to the next,
observing to write down the full sum of the left hand column.
METHOD OF PROOF.
1. Draw a line between the first and second line* of figures
to cut qfF the first number.
2. Add the other numbers together and set their sum Under
the sum of all the number*.
ARITHMETIC*. n
3% Add the numbers last found with the numbers cut off, and
if their sum be the same as that of the first addition, the work
is right.
EXAMPLES.
CO «• < 0>
4 9643 . S 2 1 4 4 3214
645 3 2 13100 1960
98760 19264 420
14000 83400 40
Sum 226935 14 7 778 45634 Sum
17. 7292115764 2420
Proof 2 2 69 3.5 147 778 4563 4 Proof
4. Add 86S5, 2194, 7421, 5063, 2194, and 1245. An*. 26754.
5. How many days in the 12 calender months? An*. 365.
6. From the building of Rome to the death of Alexander, was
683 years, from the death of Alexander to the Christian era, was
65 years ; how many years from the building of Rome to the
present date, 1818? Ann. 2566.
At the late census of 1810, the inhabitants of New England
were as follows.
Of District of Maine 228,705, New Hampshire 214,460,
Vermont 217,896, Massachusetts 472,040, Rhode-Island 76,931,
and of Connecticut 261,942^ what was the sum total ?
Ana. 1,471,974.
FEDERAL MONEY.
By an act of Congress, all the accounts of the United States,
the Sallaries of Officers, all Revenues, fee. are to be reckoned
in Federal Money, which mode of reckoning in point of simpli-
city is the nearest allied to vfhole numbers, as it increases like
<them in ten fold proportion.
ADDITION OF FEDERAL MONEY.
RULE. /
Write the numbers according to their value, that is, dollars
under dollars, dimes under dimes, cents under cents, mills
under mills, &c. then proceed the same as in whole numbers,
observing to place the seperatrix in the sum total directly under
the seperating ptints above.
i
I*
VULGAR
EXAMPLES.
._ ♦ ,
< l l
. w ,
f«)
s
D. C.
M.
8 D. C. M.
8
D. C.
M.
365
5 4
l
439 3 4
136
5 1
4
487
6
416 3 9
125
9
694
6 7
168 9 3 4
$00
9
»
439
6
fr
239 6
304
6
742
5 d
143 5
111
1 9
1
Ana. 272* 8 6 1406 6 9 3 877 7 1 .*/«.
4. What is the sum total of 8 $67,14, g 117,09, 837,75,
S 11,05, and 8 96,50? Ana. 8 629,53.
* 5. What is the amount of 8 10$,12* cts. 8 9,67, 45 cts. 67
cts. and 8 1 ,08 cts. f Ana. 8 1 13,99,5.
6. Received of A. B. and G. a sum of money*; A. paid
8 123,08 cts* B. paid four times as much as A. and C. paid as
much as A. together with twice as much B. Required the sum
of payments? Ana. 8 1723,12 cts.
SIMPLE SUBTRACTION.
Is the operation by which we take a lesser number frtm a
greater of the same denomination, and thereby find their differ-
ence, or remainder.
The lesser number is called the subtrahend, the greater
number the minuend, and the number found by the opera-
tion the REMAINDER, Or DIFFERENCE.
rule:
1. Place the less number under the greater, so that units may
stand under units, tens under tens, &c. and draw a line under
them.
2. Beginning at the right, take e^ch figure in the subtrahend
from the figure over it, and set the "remainder under the line.
3. If the lower figure be greater than the ©ne over it, add ten
to the upper figure, from which figure so increased, take the
lower, and write the remainder, carrying one to the next figure
in the lower line, and thus proceed till the whole is finished. *
PROOF.
Add the remainder to the less number, and if the sum be
equal to the greater, the work is right.
* If the cents are less than 10 place a cypher in the tens place,
or place of dimes; example, write 10 dots, and 3 cents, 8 10,03,
ARITHMET1CK.
*9
EXAMPLES.
From
Take
H8r 6 2$
2 3 4 3 7 5 6
; £rom
Take
Rent.
Ptoof
. r (2)
5 3 2 7 4 6 7
10 8 4 3 8
Item.
9 4 3 8 6 9
4 3 19 2 9
Proof
3 2 8 7 6 2 5
5 32 7f4i <6 7
S. From 2637804, take 2376982. sAn*. 26082?; »'
4. From 3762162, take 826541. ^n«. 2935621.
5. From 78213666, take 27821890. An*. 50391716.
~ 6. From ninety seven thousand six hundred and thirteen,
subtract thirty thousand nine hundred and nine* An*. 66704.
7. From the destruction of Carthage to the year of our Lord
1818, was 1965 years, how long before the Christian era was
the city, destroyed I An*. 147 years.
8. Gun powder was invented by a Monk of Cologne,, in 1330,
how long has4t been in use to this date 1818 ?
An*. 4&8 yean.
9. The Arabian method of Notation was first known in Eng-
land in the year 1150 ; how long thence to 1818 ?
An*. 668 years.
SUBTRACTION OF FEDERAL MONEY,
* RULB.
Place the numbers according to their value, aiid subtract as
in whole numbers.
EXAMPLES.
0)
8 D. C. M.
From 48 9 6 4
Take 37 8 9 S
(*>-"V
8 D, C. M.
From 940 6* 4 9
Take - 804 9 6 3
An*. 11 7 1
An*. - 135- 6 8 6
From 125 dols: take 9 dols. and 9 cts*. An*, g 115 91 cts.
From 127 dols. 1 cent, take 8 41 10 cts. An*. 8 85 91.
From 365 dolls. 12 cts. take $ 18T 3$ cts. An» t .t'\ff 77.
From 100 dolls, take 99 cts. An*, g 9£ 01 cent.
A. owes B. 1000 dols. and pays him in part, as follows*
at onetime 8 236 10 cts. at another 8 108 25 cts. and lets
him have fifty bushels of wheat at 2 dollars pr. bushel, together
with a horse worth 85 dols. and a plated harness valued at 8 <•>$
75 cts. 5 it is required to find how much is yet due ?
•" An*. S 409 90 cts.
o.
4.
5.
6.
7.
viz.*
14
VULGAR
SIMPLE MULTIPLICATION.
Is the operation by which we increase, or repeat One of two
numbers of the same denomination, as often as there are unks
in the other.
The number to be multiplied, is called the multiplican».
The number by which we multiply, is called the multipier.
The number found by the operation, is called the propuct.
The Multiplicand and Multiplier, arc both called factors.
RULE.
I Place the Multiplier under the Multiplicand), so that uniu
may stand under units, tens under tens, &c. and draw a line un-
der them.
2. Begin at the right, multiply each figure in the Multiple
oand by the Multiplier, carry one for every ten, and you w|Jl
have the product.
PROOF.
Multiply the Multiplier by the Multiplicand.
EXAMPLES.
0> .;..., . (2)
Multiply 436 Multiplicand. Multiply 90036
by 5 Multiplier. .by: 7
2180 Product.
(3)
Multiply 34293
by 74
137172
- 240051
. . Am v 630252
Multiply 32745654473
by' 234
Ans. .2537682
1309826.17892
98236963419
6,5491 $08946
A . ". ' „-.j"^ i ji i ■ i
Product 7662483 1466&2
L
5. Multiply 364111 by 56. . Ana. 20390216.
6. Multiply 7128368 by 96. 4m. 684323328.
7. Multiply 123456789 by 1440. Ane. 177777776160,
CONTRACTIONS.
When there are cyfihers on the right of one, or both the
factors:
RULE.
Proceed as before. Multiply by the significant figures, neg-
lecting the cyphers, and on the right of the product place as
many cyphers as were neglected in both the factors.
ARITHMETICS.. 15
s
EXAMPLES. • .'.'
1. Multiply 1234*00 (2)
by 7500 360000 by 120000a
" ■ ■ • , 1200000
61725 '• :
86415 432300000000 An:
Product. 9258750000
3. Multiply 461200 by 72000. Ana. 33206400*0*,.
Multiply 815036000 by 70300 Am. 57297030800000.
MULTIPLICATION OF FEDERAL MONEY.
RULE.
Multiply the given price by the quantity) and carry as in
whole numbers. The seperatrix will be as many figures from
the right hand in the product, as in the given price.
examples.
). What wHl 36 yds. of broad cloth come to at six dollars
twenty fire cents pr. yd. ?
8 6 25
36 yards.
37 50
187 5
Am. % 225 00 225 dolls. cents.
2. What cost 15 yds. calico at 67 cts. pr. yd. ? An*, g 10 0$.
3. Wbat cost SS yds. kerseymere at g 1 67 cts. pr. yd. ?
Ana. g 58 45.
4. What is the value of 75 yds. satin at 2 3 75 cts. pr. yd! ?
Ans. 8281 25.
5. What tost 131 bushels wheat at 8 I 67 cts. per bush. >
Ana. 8 218 77.
6 What cost 126 gallons molasses 32 cts. per gal. ?
An 8. g 40 96.
7. What will 66 bushels oats cost at 28 cts. pr. bush.
Ana. g!8 48.
& What cost 97 lb. sugar at 1 2* pr. lb. ? Am, g 12 12$
1
VURGAR
Mr. William Williams,
Bought of James ftfcrcb&if.
10 ibs. Green Tea, at g 2 15 cts. pr. lb.
36 » Coffee,
15 " Loaf Sugar,
o cwt. of Malaga Raisins,
36 gallons Wine,
91 " Molasses,
21*cts.pr. lb.
25 cts. pr. lb.
7 00 pr.cwt.
2 12 cts. pf. gal.
36 cts. pr. gal.
Received payment.
New-York, the 1 st May, 1 8 18.
g 136, 59
James Merchant.
SIMPLE DIVISION. .
Teaches to find how often one number is contained in another
of the same denomination; or to find a quotient which multi-
plied into the divisor will produce tho dividend.
The number to be divived, is called the Dividend. %
The number to divide byy is called the Divisor.
The number of times the dividend contains the divisor, it
called the Quotient.
RULE.
1. Assume as many figures on the left hand of the dividend as
contain the divisor once or oftener ; find how many times they
contain it, and place the answer on the right for the' first figure
of the quotient.
2. Multiply the divisor by the figure you have found, and
place the product under that part of the dividend from which
it was obtained.
3. Subtract the product from the figures above it ; then bring
down the next figure of the dividend and place it at the right
handsof the remainder ; divide the number it makes up as be-
fore, and proceed in this manner until the whole is finished. '
ARITHMETIC*,
If
How many times arc 5
contained in 137906 ?
PROOF.
Multiply the divisor into the
quotient, add the remainder if
there be any, to the product ;
if the work is right the sum
will be equal to the dividend.
&XAMFLE3,
Dividend, Quotien
Divisor 5) 137906 (27581
> - 10 5
37 proof. 137906
35
29
25
40
40
6
5
Divide 33489 by 9,
Divisor, Dividend.
9) 33489 (3721 Ana.
27 9
64 3S489 proof.
63
18
18
1 remainder.
Divide 11680 by 32
32)11680(365 Ana*
96
208
192
160
160
— * Proof by addition 1 1680
9
9
Divide 1893312 by 912, Ana. 2076.
Divide 1893312 by 2076, Ana. 912.
How often does 761858465 contain 90001 ? Ana. 8465.
Divide 280208122081 by 912314. Ana. 30714O TT Vrrr
CONTRACTIONS.
To divide when there are cyfihera at the right hand of the
diviaor.
RULE.
Cut off the cyphers from the divisor, and just the same num-
ber of digits from the right of the dividend ; then divide the
remaining figures as usual, the quotient will be the answer.—
* Add the remainder and all the products of the several quo-
tient figures (multipled by the divisor) and the sum, if the w«(k
be right, will be equal to the dividend.
3
IS
VULGAR
To the remainder (if there be any) annex those figures cut off
from the dividend* and you will have the true remainder.
EXAMPLES.
(0
Divide 460000 by 1200
12(00)4600)00(383 Ana.
36
(2)
Divide 7600 by 40
4(0)760)0(190 Ana,
4
100
36
96
36
40
a
36
400 true remainder.
3. Divide 7380964 by 23000.
4. Divide 11659112 by 890000.
5 Divide 9187642 by 9170000.
6. Divide 29628754963 by 35000.
Ana. 390*9t&
Ana. 131„iiw.
Ana. l?l$Uhr
Ana. 846535^f|4f
SHORT DIVISION.
RULE.
Find how many times the divisor is contained in the fir*
figure or figures of the dividend, place the result under, and
carry as many tens to the next figure as there are ones over.
EXAMPLES.
Dividend (2)
1. Divisor 3)76432* 4)1134152
t 254776—1 rem.
283538
(3)
5)649871923
6)1027182341
00
')25OOti321702
(6)
8)11297653009
)8701 256620
10)1097654321
ARITHMETICS
11)3076259862
(10)
12)175634589
I*.
SECONDLY.
When the divisor is a composite number, or the product of
two or more numbers in the table.
RULE.
Divide successively by the component parts of the given
divisor.
EXAMPLES.
1. Divide 9125 by 25.
5X5=25; the component parts of the given divisor
then, are 5 and 5. Thus,
5)9125
5)1825
36:
5 quotient \
2. Divide
178464
by
16.
Am. HI 54
3. Divide
79638
by
36.*
Am. 2212^
4. Divide
9S7387
by
54.
Am. 17359^
5. Divide
93975
by
84.
Am. 1U8&
6. Divide
145260
by
108.
Ana. 1 345
7. Divide
1575360
by
144.
Am. 10940
To Divide by 10, 100, 1000, 10000, &c.
RULE.
Cut off so many figures from the right ©f the dividend, a*
there are cyphers in the divisor ;— that part cut off! from the
dividend is the remainder, the other figures in the dividend are
the quotient.
EXAMPLES.
(0
Divide 600065 by 1000
1(000)600)065(600 quotient, and 65 remainder.
2. Qmde- -' 65 by 10. Am. 36 and 5 rem.
3. Divide 5762 by 100. Am. 57 and 62 rem.
4. Divide 90764 by 1000. Am. 90 and 764 rem.
5. Divide 876432 by 10000. Am. 87 and 6432 rem.
* The total remainder is found by multiplying the last re-
mainder by the first divisor, and adding in the first remainder*
30 VULGAR
SUPPLEMENT TO MULTIPLICATION.
To multiply by a mixed number, that is a whole number and
a fraction.
RULE.
Multiply by the whole number, and take f , £, |, Sec. of the
multiplicand, and add it to the product.
EXAMPLES.
1. Multiply 43 by 12$
32
Ana. 5374
. ( 2 )
Multiply |)24
by Si
■ ■ «»
i)12=i
6=4
72
Ana,
90
f
An*.
20533-f
Ana*
334134
Ana.
1191418
Ana.
6598|
3. Multiply 2464 by 8£
4. Multiply 6497 by 5f
5. Multiply 12248 by 9|
6. Multiply 345 by 19-J.
Practical Que at ions in Multiplication and Division.
1. In 36 pieces of broad cloth each containing 24£ yds.
how many yds. ? Ana. 873
2. What is the product of 430 multiplied into itself ?
Ana. 184900
3. What number multiplied by 9 will make 225 ?
Ana. 25
4. What cost 9 yds. cloth at g7 pr. yd ? Ana. g 63
5. If a man spend 8600 pr. year, what is that pr. calender
month ? Ana. 8 50
6. Sold a ships cargo for 87940, required to find 1-4 of the
amount ? • Ana. 8 1985
7. The quotient of a certain number is 1 1940, and the divisor
20) What the dividend ? Ana. 238800
8. How many feet are there in a mile, or 320 rods, allowing
eath rod contains 16f feet ? Ana. 5280
9. How many yards in a mile, if $* yds. make one rod ?
Ana. 1760
10. How many yards of broadcloth, at 87£ per. yd. may be
• bought &r 37 yds. of do. at 8 S| per. yrd. ? Ana. 18£ yrds.
ARITHMETICS 31
1 1 . How much wine, at 1 J doll. pr. gal. with molasses, at £doU,
pr. gal. and of each an equal quantity, must be had in exchange,
for 35 gallons of brandy, atg 2,25, pr. gal. and 10 gallons of eld
spirits, at $ 1,1 2$ pr. gal. ? Am. 40 gallons.*
COMPOUND ADDITION.
Teaches to unite several numbers of different denominations—
as pounds, shllings, pence, &c. into one sum.
RULE.*
1. Place numbers of the same denomination under each
other.
2. Add the figures in the right hand column, and find how
many of the next denomination are contained in the sum,
which carry to the next denomination-, observing to set down
the remainder under the column added, and thus proceed with
all the columns excepting the last, where the whole sum is t»
fee written down.
STERLING MONEY.
£ . *. d. . £. *. (/. yr. £. *. d. gr,
17 13 4 84 17 5 2 47 13 6 O
13 10 2 75 13 4 1 19 2 9 2
10 17 3 51 17 8 3 14 10 11 1
& 8 7 20 10 m 1 12 9 i 3
3 3 4 17 15 4 2 8 7 6 2
8 8 , 10 10 11 12 7 t
54
54 1
261
5
8
1
176
8
2
3
261
5
8
1
* The reason of this rule will be obvious, if we consider, that
1 in the column of pence, is equal to 4 in the column of forth*
ings, and 1 in that of shillings; to 12 in the column of pence fee.
V?- VULGAR
•(*)• ■: w ■ (6)
£. 9. d. gr. £ . *. rf. gr. £. 9. d. yiv
II 9 6 1 144 9 12 I 987 6 9 1
19 8 4 • '»• 160 19 10 3 17 19 11 2
99 11 10 2 140 4 2 6 4 3
6 O 4 1 910 4 3 89 6 10 2
2. TROY WEIGHT.
lb.
or. /iwf. $>r.
lb.
OZn flXOt.
gr.
lb.
oz. /l«tf .
gr*
19
11 19 23
11
9 6
4
9
11 19
23
Jl
10 11 20
20
7 3
20
8
10 18
22
6
9 17 10
10
9 16
17
7
9 17
21
4
8 4 9
9
8 14
23
19
8 16
20
3
7 12 17
17
4 19
3
18
7 15
19
3. AVOIRDUPOIS WEIGHT.
wt.
yf.
/*.
lb.
oz.
dr.
r.
C*tff.
gr.
lb.
oz.
2
3
27
25
13
15
90
4
2
17
14
4
2
18
24
10
14
100
.4
3
27'
15
6
1
17
23
11
13
86
19
2
19
12
9
16
18
9
10
H
13
1
•
3
«
14
27
7
8
96
10
3
17
14
4. CLOTH MEASURE.
Y<L
gr.
na.
JS.£.
gr.
na.
£.F.
gr.
na,
70
3
3
44
2
2
90
SL
3
13
2
1
60
1
3
108
\
3
90
90
1
3
76
3
2
8
1
3
30
3
3
40
1
3
6
3
1
20
3
95
2
5. DRY measure:
fik. gt. fit. bu. file. gt. bu. fik. gt. fit.
17 70 3 4 95 2 1 1
2
6
1
60
2
1
76
2 .
.7
3
5.
1
55
3
7
40
2
6
1
1
4
60
1
6
26
2
4
2
* 1
9
1
6
3
2
5
I
ARITHMETICS <%s
9 WINE MEASURE.
tal.
f*
fit.
hhd.
gal.
gt.
fit.
fiiflc
hatib
psl
39
3
1
4*
61
3
1 .
34
2
2
36
3
20
35
2
11
1
3
35
2
1
ifr
24
1
1
7
3
2
32
3
17
U
3
19
1
3
11
2
1
10
9
1
45
1
,1
7. LQNG MEASURE. .
Yds.
ft.
in. be.
m.
fur.
fiol.
le. m.
fur.
fioL
4
2
11 2
46
3
16
85 2
7
27
1
2
9 2
91
1
29
75 2
5
19
2
1
8 1
67
3
IS
25 1
4
23
1
7 I
60
7
33
95 1
6
11
3
1
10 1
35
2
11
11 2
3
15
1
*. LAND,
OR SQUARE MEASURE
teres
roods
rocf«
acre*
\ roods rods
acre*
roods
rodi
440
3
37
11
3.
17
990
3
39
760
2
33
97
2
16
760
14
600
1
14
20
3
la
17
3
38
976
2
35
36
24
32
2
20
57
20
25
8
203
3
34
9. SOLID MEASURE.
r. y*.
cord*.
ft.
/'*
inches-.
40 90
. 3
127
20
1440
203 7
20
220
26
1259
•%
23 23
35
29
22
1440
__ —
10.
TIME.
Yrs> mo.
w-
dye.
;
Yrs.
<fy«.
h.
m.
sec.
57 7
3
26
22
300
23
59
34
230 6
2
5
33
327
23
44
43
3 10
4
30
2*8"
364
20
43
58
19 9
3
9
34
303
23
34
33
J
94
r
VULGAR ..
•
11,
CIRCULAR MOTION.
#
&.
o
\
W
S.
o
.» \*
S. «
v .
t*
3
29
17
14
11
29
50 40
17
13
50
2
23
57
44
10
20
30 45
34 25*
49
35
4
22
20
40 •
37
18
48 29
42 10.
38
r
1
30
S3
40
23
17
59 57
20 12
36
54
4
14
45
55
99
29
40 50
44 20
33
6
24
42
8
17
39 42
5 27
15
42
COMPOUND SUBTRACTION,
Teaches to find the difference between any two sums of dif-
ferent denominations.
RULE.
Place numbers of the same denomination under each other,
the less below the greater ; begin with the least denomination,
and if it exceed the figure^ above it, borrow as many units as
make one of the next greater ; subtract it therefrom, add the up-
per figure to the difference ; always adding one to the next
higher denomination for that which you borrowed. JPjROOi^-—
the same as in Simple Subtraction.
1. STERLING MONEY.
EXAMPLES.
£ . *. d. qr. £. s, d. qt.
From 346 16 5 3 494 17 9 3
Take 128 17 4 2 479 19 10 2
Bifference 217 1? 1 1
lb. oz. fiwt.
12 10 19
9 11 18
2. TROY WEIGHT.
oz fiut. gr*
10 19 23
6 17 19
lb. oz. fiwt. gr.
120 10 16 19
134 11 15 17
arithmetics:. js
8. avoirdupois weight-
ib. *z. lit. twt. qr. lb. T. cvt. qt. fa ox. <Jr.
9 r 15 7 3 13 3 10 3 54 IS 1$
4 9 26 5 2 15 2 12 2 29 14 15
74.
13
3 $
% I
tfik. qt.
1 1
2
?,al.
39
36
?'. /^r. gi.
3 12
3 O
4. CLOTH MEASURE.
JB.Ji.-fr. na. E.JP. qr. «*«
64 2 2 190 2 B
40 13 08 1 3
8. DRY MEASURE.
bu. fyk. qt. bu. ilk. qU
70 3 4 95 2 1
60 2 1 76 2 7
6 WINE MEASURE.
A&eL g?L $$* jOr hhd. gal. qt. fit,
261 3 12 34 2 2 1
.- 1 ,■ 1 'U.H J f .M ■ I l lu ll. W l l ■ — IMp, '
Yds. ft. in. be. m. fur* flol. le. m* fur. fiol*
,4 %, U ,2 -.. ?ft $, - i$., . S5 2 7 27
' 1" %; . 9 V 2 \ , 4\ ..■ i . 2* . . 75. ,,$ * • 19 -
*. LAN©, OR SQUARE MEASURE.
vfcri* r**rf* ro<& , ech;* rood* rorft \feet. inches
760 2 38 97 2 16 960 14
440 3 .37 ... H 3 17 790 33
m
VULGAfc
V. SOLID MEASURE.
3P. ft* <#rd%. ft. r T. ft., inched
240 90 3 127 1440 20 238
03 7 2 220 ' 26 18 135
fr*.
230
57
7
>6
3 26
~2 5
10. TIME.
Yt*. dya. h. m; sec.
32 300 2S 59 34
23 327 23 44 43
11. CIRCULAR MOTION.
3 29 17 14
2 ?3 57 44
H 29 50 40
10 20 30 45
S - o \ M
37 13 69
14 25 49 35
* . -
COMPOUND MULTIPLICATION.
Shows how to find the amount of any given number of diverse
denominations, by repeating it any proposed number of times.
FEDERAL MONEY.
-;, RULE.
Multiply as in whole numbers ; and place the *eperatrix as
many figures from the right hand in the product, as it is in the
multiplicand. v ~
EXAMPLES.
8 cts.
Multiply 17 18 by 25
25
>h G. C. I
doh It. c. In.
Multiply 7 13 9
14
by 14
85 90
343 6
Preduct. g 428, 50
> 28 5 5 6
71 3 9
Product. $ 99, 9 4 6
3. Multiply 1 1 mills by 40 Ana. 44
4. Multiply 41 cents 5 mills, by 150 Am. 62 25
ARITHMETKX. "&
S. Multiply * dols. by 50 fna* 450 00
^ 6. The number of inhabitants in the United States is 7 mil*
lions ; if each should pay the sum of 8 eents yearly, for 9 year*,
how many dollars would be raised I Ana. 5 millions.
WEIGHT^ MEASURE, STERLIJYG MOJVEYy&c.
RULE.
Write the multiplicand, and place the quantity under. the least
denomination for the multiplier? observe- the same rules, for car-
lying as in compound addition.
1. What cost 9 lb. of* sugar* at 2*t 8<& 2gr. pr.lb, I
2* 8$rf ,
Ana. XL 4* 4%d
2. 3 lb. of green tea, at 9*. 6d. pr. lb. ? An*. £ \ , 8* Sd
3. $ lb. of loaf sugar> as 1 *. . 3d. pr, lb r ? Ana. £0 '6,3
4. 9 cwt. of cheese, at 1/. 11*. Sd. pr. cwt. Ana. 14/ 2a 9d
When the multiplier excteda 12.;
Multiply successively by its component parts* instead of the
whole number.
Examples.
L 16 cwt of sugar, at XL 18*. 8d. pr. cwt.
(1) (2)
1/ 18* %d 28 yds.. of broad cloth, at 19*. 4d.
4 J»r.yd. Ana. £27 ', Xa.4d.
7 14 8
4
£ 30 18* Sd
3. 182 yds. Irish linen, at 2*. 4d. pr. yd. v Ana. #15, *»
*4w 144 reams of paper, at 13*. 4d. pr. ream. Ana. £ 96,
5.^96 bushels wheat, at l/. 3*. 4d. pr. bush. Ana. £ 112,
If no two numbers multiplied together will exactly equal the
multiplier, multiply by any two numbers that come the nearest,
then multiply the upper line by the remainder, which added to
the last product gives the answer.
f VULGAR
I. What will 47 yds* hroad cloth come to, at Its. *<t. pr.jd. ?
£ ** &
0, 17, 9 price of 1 yd.
5
4
8
9
9
39
18
9
1
15
6,
41,
14
3
price of 5 yds.
price Of 45 ft*
price of 2 yds.
Ana. 41, 14 3 " price of 47 yards*
2. 29 yds.of cambricd,at 0/. 13*. ?d. pr. yd.
wfiu.i"19, 13*. HA
3* 111 yds. broad cloth, at 1/. 2*. 6^. pr. yd. . .-.
Ant.£\ < Uy 17*. 6rf.
4, 23 ells, at X>£ 3*. 6ftf. pr. ell ? 4n*.£4> U* sd.\
8. 117 cwu Malaga raisins, at £ly 2*. 3rf. pr. cwt.
An*.£lj>0 9 $a* 3<f.
6. 59 yds., tabby velvet, at 7s. lOd. pr. yd. Arts. £ 23, 2*. 2tf.
7. What is the weight of 7 hhds. of sugar, each weighing
9 cWt. 3 qr. 12 lb. I An*. 69 cwt.
8. In 9 fields, each containing 14 Acres, 1 rood, and 25 pole%
bow many acres \ Ana. 1 29a. iqr. 25rods.
9. In 6 parcels of wood, each containing 5 cords and 96 feet,
bow many cords ? Ana. 34-J cords.
& * k \\ . " &• ° v vv
10. Multiply 1 15 48 24 by 5. Arts. 7 19 2 O
ill. Multiply 3 cords, 87 feet, by $ Ahs* 29 cords 56 feet.
COMPOUND DIVISION.
Teaehes to find how often one number may be had in another
of different denominations.
ROLE*
Write down the given sum in cents, and divide as in %hele
lumbers ;— the qHotient will be the answer in cents.
* Note. If the cents in the given sum are less than l6, place
a cypher on their left, or in the ten's place.
ARITHMETIC*. %?
EXAMPLES.
Divide 8 674 19 cents by 24
(1) dol. cts. m.
24}67419(28, 0$ 1J
48
To bring cents into 1 94 Wh,en there is a remainder,
dollars, you need only 192 add a cypher, again divide*
point off 2 figures on - — and you will have the mills,
the right, the rest 219
will be dollars. 216
3a
24
2' Divide 4 dols. 9 cents, or 409 cents by 6. Ana. ,68, cts*
3- Divide 9 dols. 24 cents, by 12 % Ana. ,77 cts.
4 Divide 2 dollars into 33 equal parts/ Ana; ,66, cts,
5. Divide 1000 cents by 25. Ana. j40 cts.
6. Divide 999 cents; by 9. Ana. Ill cts.
7. Divide 1 <!<>!. by S mills. Ana. ,500 mill*
8. Divide 125 dols. by 500. Ana. 25 cts;
9. Divide 10 cents, by 10 mills* Ana. 10 mills.
STERLING MONEY, WEIGHTS, MEASURES, &c.
UULE.
Begin at the left hand, as in simple division, and if any thing
remains, determine how many of the next denomination the .re-
mainder is equal to, which add to the next denomination, con-
tinuing to divide, and to carry the remainder, as before, till the
whole is finished. (1)
£. a. 'd. M
Pivide 19 4 10 3 by 4,
£.
8.
d.
?r,
19
4
10
3
4)19
4
10
3
£.4 16 2 2 | Ana.
£. a. d. 9 r.
2. Divide 31/. 14*. 9d. 2gr. by 17. Ana. 117 4
3. Divide 119/. 12*. 2d. 3gr. by 9. Ana. 13 5 9 2
4. Divide 1/. 19*. Bd. Ogr. by 1 1. Ana. 3 7 f
Note. When the divisor exceeds 13, and is the product of 2
numbers ; divide by one of these numbers first* and the quotient
by the other, the last quotient will be the answer.
30 VULGAR
5. Divide 128*. Sii.Od. Oqr. by 42'. Am. 3 S 1 Q !►
6. Divide 5/. 10*. id. Oqr. by 81. Am. 14 £
7. Divide 6 tons, 1 1 cwt 5 q r »- 19jl >- bv *•
^>w # I r. 12 rwf. Syn 25#. ISoau
8. A. piece of cloth containing 24 yds. cost £. 18. 6*. what is.
it pr. yd. ? Ans* 15*. 3d
9. Divide IT Ida. 1 m. 4 far. 21 pols. by 21..
Am. 2 m. 4 fur. 1 pol.
10. From a £iece of cloth containing 64 yds. and 2 nails, a.
aylor was directed to make 27 coats ; what did each coat coHr*
lain f Ana. 2 yds. 1 qr. 2 na.
REDUCTION.
Teaches to change numbers from one denomination to aneth-
<sr, without altering their vahie.
Reduction is either Ascending, or Descending. It is Ascen-
ding, when numbers of a lower denomination are raised to a~
higher denomination. It is Descending, when numbers of a
higher denomination are reduced to a lower denomination.
PROOF. Invert the order of the question.
REDUCTION DESCENDING.
RULE.
Multiply the highest denomination given, by so many 'oS the
next less, as make one of that greater ; and thus continue til£
you have reduced it as low as the question requires.
EXAMPLES.
!. In 37/. 13*. Id Zqr. hew many farthings. I
20
753 Shillings.* 4)3617*
12 —
$043 pence. PROOF*
12)9043 -f 3qr r
I 20)75,3 -f- 7d.
As. 36175 farthings^ £37,13,7,3
2. In 23/ lit* 7\i. how many farthings ? Am. 22638.
3. In 47/. 19*. 3d. how many shillings, pence and farthings £
Am. 959*. 1 15 lid and 46044?r.
♦Note. In multiplying by 20, add in the shillings, by 12 add
ia the pence, and by 4, add in the farthings, if any, in all similar
&S8ft.
ARITHMETIC*. *l
4. In 315 dolls. 50 cenu, how -many threepences and farth*
Ings ? ./fa*. 7572 threepences, 90864 qr.
5. In 121 French crpwns* at 6*. 8c?. each, how many pence and
ifcrthings? An*. 9680d. 38720 qr.
6. In 312/8*. 6id. how mariy sixpences and balf-pences ?
An*. 12497 sixpence*, 149964 half pences.
REDUCTION ASCENDING.
RULE.
Divide the lowest denomination given, by se many of that de-
nomination, as make one of the next higher, and so on through
all the denominations, as far as the question requires.
FRO OF. Multiply inversely by the several divisors,
examples: ;,
1. In 122318 farthings, how many pence, shillings, and
founds ?
Farthings in a penny = 4) 1 223 1 8.
Perice in a shilling =» ; 12)30579 -f- 2qr.
. Shillings in a pound =: 2,0)254,8 -J- 3d,
An*. £127,8s.3d.2qrs.
2. In 30329 farthings, how many pounds \ , , .
AH*. £ n t lU lOrf. Iqf.
3. In 46044 farthings, hofr many pence, shillings, and pounds?
- ■' AH*}t\$il&. 9594.47*.
r 4. In 90864 farthings, how many dollars ? Ana. % 315*.
• 5. la 20160 pence, how many pounds ? An*. £ 84 '
2. TROY WEIGHT.
1. In a dozen of silver spoons weighing lib. 3oz. 1 Ipwt. how.
many grains ? Ana. 7464 grains.
2. In 10 ingots df gold,«ach weighing 9oz. 5pwt. how many
grains? An*. 44400
. 3. How many table spoons weighing 23pwt. each, and tea
spoons 4pwt« 6 grs. each, an4 an equal number of each sort, can
be made from 41b. loz. ipwt. of silver. An*. 36 A ;
REDUCTION ASCENDING AND DESCENDING.
3. AVOIRDUPOIS WEIGHT.
1. In 19 lb. 14oz. 1 1 dr. how many drams ?- -Ana.^ 5099
3. In 5 tons, how many drams? Am. 3867200
3. Bring 5099 drams into pounds. An*. • 19/a, 14oz. 11 dr.
4. A merchant has 5 hhds. of tobacco, each 8 cwt. 3 qrs. 141b.
and wishes to put it into boxes containing 701b. each, how many
bcrxes are requisite ? Ans r 71
3* VULGAR
-■$: CLOTH MEASURE.
1< In 5469 nails, how many yards ? Ans. 341 yds. $ qr. \ ttel
• *. In 1320 nails, how many Ells English i Ans. 61*
3. In 28 Ells Flemish^ how many quarters ajDd nails ?
•in*. 84 qr. 336 na.
4. How: many coats containing 1| yds. each, can he made
from 75 J- yards of broad cloth ? Ans* 42*
5. DRY MEASURE.
1. In 63 bushels, how many peaks, quarts, and pints ? \
Ans. 272 pecks, 2176 qts. 4352 pt$» :
2. In 25 bush. 3 pks. 7qts. how many quarts ? Ans.'szi*
3. In 8704 pints, how many bushels ? Am. 136.
4. A gentleman has 1003 bush. 3 pks. of grain, and a teayra
that consumes 2| bu^h. pr. day, how long will the grain last ?
An*, one jre^r
9. WINE MEASURE, . , , >
1. In 9* tons of wine, how many quarts ? Ans. 9072.
2. In 18144 pints of wine, hpw many hhds.? An ft* 56.
3. How many bottles containing If pint, can be filled from a
pipe of wine? ^ Ans. 672.
r. LONG MEASURE. ; r .
1, In 1 7. miles, how many inches ? Ans. 1077120.
X In £. leagues, how many yards? An*. ls5R4Q« ?
3. How, many revolutions do the forward wheels of a stage
describe in running from Hartford to New-Heven* it bfipg ?4i
miles, allowing the wheels to be 14| feet in t circumference ?.
x Ans. 12114VVT
4. What is the circumference of the globe in inches, it being
$60 degrees? Ana* 1 $«$?#• ?<>9
M ' '.'.-. /■ !'■ 8. TIME. . ... ....,.,
i. In 1 year, or 363d. 5h. 48^ 58>\jiow many second* ? 4 >i .^
Ans. 3155MW
%- In 655989 days, bow many years, reqHonia^ th* ,y$ajr to
contain 365d. 6h« ? A»*i* \%$&
' 3. How many minutes were there from the fcrith frfCtate Un'
the year 1776, allowing the length of the year the same as in the
first example? ■•••../- Aas. ?34p$5364 v 4$*
....■*'• ■ ■ - . i * •■.*'.» ■ * ' * • • • * * * ■ ..*
9. CIRCULAR MOTION, f: 7 \
1. In 9 signs, 13° 25* how many seconds? Ans. 1Q20PQQ
2. In 811480 seconds, how many signs?
Ans, 7§. U o M\40W
3. How many minutes in the whole of th* Earth's orbit, or
12 signs ? An?. 4 1.60^
ARITHMETIC*. S3
N
N
OF CURRENCIES.
HULR*
Divide the given sum reduced to shillings; to sixpences, or
pence , by the number of shillings, or pence, in a dollar, in
each state.
1. Reduce £6S 15s. New England currency to Federal Moi\*
ey. Ans. g 212 50 cts.
2. Reduce £"481 New York currency to Federal Money.
Ans. g 1077 50
3. Reduce ^"37 10s. Pennsylvania currency to Federal Mon-
ey. Ans. g 100.
To change Federal Money to the currency of each state.
RULE.
Multiply the given sum in cents, by the number of pence in a
dollar, and cut off two figures to the right of the product, what
is left will be the answer in pence, and if the figures thus sepe-
rated, be multiplied by 4 ; and 2 figures again cut off as before,
those at the left hand will be farthings.
EXAMPLES.
Reduce g 438 42 cts. to New- York currency.
Ans. £\75 7s. 4£d^
Reduce g 1971 96 cts. to New-England currency.
' Ans. £591 lis. 9d.
Reduce 85 dolls. 43 cts. to sterling money.
Ans. £19 4s. ad.
Note — When the given sum is dollars, multiply by the nuitt*
her of shillings in a dollar.
54 VULGAR
«j ?. •sOTr» OOC0OKO " N oow 4.
2J S £ I .oo«o*o^.o © a m o T
•2 -S 5 -suwol « 000 > c> ' 0,£ > ,s < ~ o <* <* £
£ £> -s^noai-fiS^S^^^^^ --oof
g South-Carolina 'p'ooooo.«oo«
•O and
(£i Georgia.
f^00©00>*00<©
% ^ Q O O » — — 00 K
§ New-Jersey, Delaware rtioo<ooo<ooo co <© co to ^
to and 4 »ao2«22»'N « ** - - A
S Maryland. l^o « ^ w - -< - - ooooT
g 1 I • ,i, mm _ ^
52 XT— « V~«U /"^lOO. O O Q O O O ft O rt» K. T
55
New-York
and
•g. South-Carolina,
•12
r^oooooooo o»oo»**-T
J (q00 ^ <OC0N «A 00 00 oo ^ ^ A
t^)<0 Wujn-*hh^ OOO O JL
~ Vermont, New-Hamp- p*ooooo<oo.o «©*<"£
.ti shire, Massachusetts, \ .<ococo«>co**c*c* <o to - ~ ¥
£ Rhade-Islatid Coanecti- ] - ^ ^
*cut and Virginia. (Js*-* $c *•-• — — — : r« oooof
SB i .. i . . . i ■ - — — t
|j . Sterling money ^oooooooo o <o o o ^
T3
of
«> Great-Britain.
l ^SS^^^^SIS .vj ^. -« o- A
*-*s*co — o — -« — o o o o o o ±:
52 Standing weights. - S«£coo»«oo»o»«m««p ooseowt
a U— ~ —H—l
§ * s s| « i
3 .• -S -8.9 So a
**« S £BQS meet* ^Q c 0h
ARITHMETIC*. SS
DUODECIMALS.
Duodecimals are so called, because they decrease by twelves,.
from the place of feet towards the right. Inches are sometimes
called firimesy and marked thus * ; the next division, after inches,
is called parts, or seconds^ and is marked thus " ; the next is
third*) marked thus "' ; 8cc.
Multi/ilicatien of Duodecimals ; or Cross Multiplication.
RULE.
1 . Under the multiplicand write the same denominations of
the multiplier, that is, feet under feet, inches under inches, Ssc.
2. Multiply each term in the multiplicand, beginning at the
lowest, by the feet in the multiplier, and write each result un-
der its respective term, observing to carry an unite for every 1 2 y
from each lower denomination, to its next superiour.
3. In the same manner, multiply every term in the multipli-
cand by the inches in the multiplier, and set the lesultof each
term one place farther toward the right of those in the multi-
plicand.
4. Proceed in like irmnner with the seconds and all the rest of
the denominations, if there be any more ; and the sum of all the
several products, will be the product required.
The products of the several denominations depend upon the
principle, that—
Feet by feet give feet. Primes by primes give seconds.
Feet by primes give primes. Primes by seconds give thirds.
Feet by seconds give seconds. Primes by thirds give fourths.
&c. Sec.
Seconds by seconds give fourths. Thirds by thirds give sixths.
Seconds by thirds give fifths. Thirds by fourths give sevenths.
Seconds by fourths give sixths. Thirds by fifths give eights.
&c. Sec.
Or, in general
When feet are concerned, the product will be of the same de-
nomination as the term by which the feet are multiplied. When
.feet are not concerned, the name of the product will be express-
ed by the *um of the strokes, or 7:iarks over both the factors.
3* VULGAR
EXAMPLES.
I. Required the contents of a ceiling 10ft- 4* 5" by 7ft. 8* 6' 3
ft. * "
10 4 5
7 8 6
72 6 11
6 10 11 4
5 2 2 «
Feet 79 IT 0" 6 m 6"" Ans.
2. How many square feet in a board 17 feet 7 inches long,
and 1 foot 5 inches wide? ^ Ans. 24f. 10' 11"
3. How many solid feet in a stick of timber 12 feet 10 inches
long, 1 foot 7 inches wide, and 1 foot 9 inches thick ?
Ans. 35f. 6' 8" 6*"
4. Required the number of solid feet in a load of wood, that
is 9$ feet long, 3-} feet wide, and 3 feet 7 inches high I
Ans. 1 13f. 5* 8"
5. How many yards of painting in a room 20 feet in length,
14t feet in breadth, and lOjfeet in height, deducting a fire place
of 4f. by 4^-f. and 2 windows, each 6f. by 3£f. ? Ans. 73 T * T yards.
* In like manner may pounds, shillings, pence, 8cc. be multl*
plied into each other, by observing the following principle—-
that Pounds by pounds give pounds.
Pounds by shillings give shillings.
Pounds by pence give pence*
8cc.
Shillings by shillings, every 20 is 1 shilling, every 5 is Sd.
and each 1 is 2 farthings, and 4 tenths of a farthing.
Shillings by pence, every 5 is a farthing* and each one 2
tenths of a farthing, &c.
Pence by pence, every 60 is a farthing, and every 6 one tenth
ef a farthing.
^ EXAMPtES.
1. Let it be required to multiply 31. 5s. 6d. by 21. 42s 9d*
£ a. rf,
3 5 6
2 12 9 . ^
6 110 qrs. tenths.
1 19 3 2 4
2 5 19 '
Ans. Sl.ias.Qd.Oqr,^
ARITHMETICS Sf
2. Let it be required to multiply 2s. 6d. by itself.
^ Ans. 3d. 3qta.*
In the above example, a fiound was considered the integer ;
but when a shilling is taken for the integer, observe the follow-
ing precepts ; — namely, that
Shillings by shillings give shillings.
Shillings by pence give pence.
Shillings by farthings give farthings, Sec.
Pence by pence, every 1 2 Is a penny, and each 3 a farthing.
Pence by farthings, every 12 is a farthing, and each 3 is £ of a
farthing, &c.
Farthings by farthings, each J 2 is £ of a far thin gr
examples
1. Let it be required to multiply 2s. 6d. by 2s. 6d. one
shilling being taken for the integer.
s. d» (2)
2 6 Multiply 2\ feet by 2* feet.
2 6 2f. 6'
2 6
5
1 3 o 5
— — 13
£0 6
Feet 6 3' Ans.
Where it is apparent, that if, instead of shillings, pence and
farthings, we reckon feet inches and quarters, the result will
be the same.
Xhe two following questions are Sexcessimals.
3. If two places differ in longitude 2° 12' j what is their dif-
ference of time ?
, Mult. 2° 12* 00". 00"' .
by S* 59" 20"* the time in which the Sun passes
• — — — through one degree.
Ans. 8* 46" 32'"
4. Two places differ in longitude 31° 27' 30", what is the
difference in time of the Sun's coming to the meridian of those
places, the Sun passing through 15° in an hour I
31° 27* 30"
4* 00" In 4 minutes of a solar day, the Sun
passses 1 degree.
Ans. 1° 6* 30" 00"*
"* Note. Whence it is manifest that fractions multiplied,
become less, in the same proportion as integers^ by multiplying,
become greater.
■'•3* VULGAR
SIMPLE PROPORTION, OR RULE OF THREE.
Teaches, that by having three proportionals given, to find a
fourth, which multiplied into the first, shall be equal to the pro*
duct of the other two: or the rule of three teaches, by-
having three numbers given, to find a fourth, which shall have
to the second the same ratio, that the third has to the first.
Proportion in common Arithmetics, is generally consid-
ered direct, or inverse. It is direct, when more requires more,
or less requires less, and inverse, when more requires less, and
less requires more.
1. Observe that two of the given numbers in the question are
always of the same kind, one of which must be the first number
in stating, the other the third ; consequently that number which
is of the same kind with the answer or thing sought, will always
be the second, or middle term in stating.
2. Observe farther ; the third term is always a demand, and
may be known by its asking the question.
RULE.
1 . Write the numbers so that the term which asks the ques-
tion may possess the third place, and that which is of the same
kind with it, the first place, the remaining term will posses
the second place.
2. Bring the first and third terms to the same denomination,
and reduce the second, to the lowest name mentioned in it.
3. Multiply the second and third terms together, and divide
jtheir product by the first, the quotient will be the answer in the
same denomination as the second term.* '
* This rule is founded on the principle that if four numbers
be proportional, the product x>( the extremes, is equal to the
product of the means.
Thus 4: 8:; 16:52, here 4X32 = 128 ; and 8 X 16= 128 :
whence it is evident, that dividing the product of the means^ or
the product of the extremes by the Jirst extreme, the other ex-
treme is obtained. Thus— dividing 128 the product of the
means, by 4, gives 32 for the other extreme ; and dividing the
product of the meuns, or the product of the extremes by one of
the mean*, the other mean is obtained. Thus— -dividing 128 the
product of the extremes, by 8, gives 16 for the ether mean ;
hence the propriety of the rule, in multiplying together the 2d
and 3d terms, or the means, and dividing by the first term, or the
first extreme, to obtain the other ; for it has been demonstrated,
that the product of the means, divided by the first * extreme*
gives the other extreme.
ARITHMETICS 3*
This- rule is applicable when the proportion is direct ; but
when the proportion is inverse j that is, when the conditions ef
the question require the answer to be greater, or less than the
second term : Multiply the first and second terms together, and
divide the product by the third. The quotient will be the an-
swer in the same denomination as the middle term.
EXAMPLES.
1. If 6 yards cost 8 18) what will 12 yards cost at the same
rate? 6 s 18 : : 12
13
6) 216 (36 dollars.
18
36
36
do
If 1 8 dollars buy 6 yards, how many yards will 16 dols. buy ?
18 ; 6 : : 36 (3)
6 If 8 36 buy 12 yds. how many
—~ will 18 dollars buy.
18)216(12 Ans. S6 ; 12 : : 18
18 " 12
36 56)216 (6 Ans;
36 216
ae ooo
40 VULGAR
i
4. If 12 gallons brandy cost 25 dols. 44 cents, what will be the
price of 2 pipes, at the same rate.
Gal. g Pipes.,
12 : 25,44 i : 2
2 hhds. in 1 pipe.
As. 12
4
63
gal. in 1 hhd.
12
24
; 2544 :
252
i
252
do. in 2 pipes
5088
12720
5088
12)631088 (525 90 6 J Ans. . five hundred and
60 twenty-five dollars, ninety
~ cents, 6 mills and two
31 thirds,
24
"to
60
108
lo8
80
72
. *
12
5. If 5 horses eat 10 bushels of corn in a week, how many will
35 eat in the same time ? Ans. 70
6. If an ounce of silver is worth 90 cents, what is a cup
worth that weighs 21b. 10 oz. Ans. g 37 80
7. If 15° of the equator revolve through the meridian in one
hour, in what time will 150° 51* 15" revolve through ?
Ans. lOh. 15* 25°
8. What is the tax upon g 50 97, at ten cents on the dollar.
Ans.- g 509 70.
9. What wilj 4 casks of raisins, weighing 2 cwt^ 2 qr. 25 lb.
ARITHMETICS 41
come to, at 16 cents pr. lb. ? Ana. £ 195 29
10. *At the rate of 15 deg. pf. hour, how much of the equator
will revolve through the meridian in 12 hours 2 min. *6 sec. ?
Aps. 179 deg. 36 min. 30 sec.
1 1. When the Sun is on the meridian of London, what o'clock
is it at Mexico North America, 100 degrees 5 min. 45 seconds. I
Ans. 5 o'clock, 19 ip. 37 sec. a. m.
12. What o*clock is it at Moscow 37 deg. 45 min. east long %
when it is noon at London ? An§. 2 o'clock 31m« p. m.
13 If the Sun comes to the meridian of London, 4h. 45m.
30 sec. sooner than it does at the Meridian of Cambridge, what
is the longitude of Cambridge ? Ans. 71 deg. 20 m. w.
14. suppose a Gentleman has an income of 8 1940 a year,
and he spends 3 dols. 46 cents pr. day, how much will he have
saved, at the years end ? Ans. 8 683 10 *
15. Sound uninterrupted, moves about 1 142 feet in a second,
1aow long then, after firing a cannon at Springfield before it will
be heard at Hartford, it being 26 miles ? Ans. 2 m.O sec.-f^T
16. In a thunder storm it was observed, that it was 6 seconds
between the lightning and thunder, at what distance was the
explosion ? And. 6S52 ft. = J^mile.
17. Suppose a rocket was seen at the instant of discharge, i2
seconds before the report, at what distance was the gun.
Ans. 2{|£ miles.
18. If 8 100 in one year gain 8 6> what will 8 314 15 cts.
gain in the same time ? Ans. 8 18 84c. 9m.
19. If 8 212 25 c. gain 8 12 37* in one year what is that pr.
cent ? Ans. 6
20. A owes B 8 1?36 59 cts. but becoming a bankrupt, he
is unable to pay more than 65 cents on the dollar, what does B
receive for the debt ? Ans. 8 H28 73c. 3^rm
*fel. If a man buy merchandize to the amount of 8 560, and
gain by the sale 8 190 40, how much will he gain by laying
out 150 at the same rate ? Ans. 8 50 00
22. If 30 men perform a piece of work in 1 1 days, how many
. men can accomplish another piece of work 4 times as large in,
a fifth part of the time ? Ans. 600
23. A wall that is to be built to the height of 27 feet, was rais-
ed 9 feet by 12 men in 6 days, how many men must be employ-
ed to finish the wall in 4 days, Working at the same rate ?
Ans. 36
24. If a stick 8 feet long, cast a shadow on level ground 12
* Note. The equator may always be supposed to revolvo
through the meridian, at the rate of 15 degrees in 1 hour of so-
lar time, without any sensible errour ; though it is a f ruction
wide of the tiuth.
6
42 VULGAR
feet, what is the width of a river, over which a tower, known te
be ISO feet in height casts its shade. Ans, 270 feet.
OF THE LEVER OR STEELYARD.
It is a principle in Mechanic ks, that the power is to the
weighty as the velocity of the weight, to the velocity of the pow-
er ; therefore to find what weight may be raised or balanced by
any given power, say ;
As the distance between the body to be raised, or balanced
and the fulcrum, or prop, is to the distance between the prop
and the point where the power is applied ; so is the power to
the weight which it will balance.
If a man weighing 1601b. rest on the end of a lever 10 feet
long, what weight will he balance on the other end, supposing
the prop 1 foot from the weight I
The distance between the weight and the prop being 1 foot,
the distance from the prop to the power is 10*-* 1 = 9feet > -
/ therefore,
Je. ft. Ih. lb.
As 1:9 : : 160 i 1440 Ans.
If a weight of 1440 be placed 1 foot from the prop, at irhat
distance from the prop must a power of 1601b. be applied to.
balance it ?
As 160 : 1440 : i 1 : 9 feet. Ans.
At what distance from a weight of 14401b. must a prop be
placed, so that a power of 1601b. applied 9 feet from the prop,
may balance it.
As 1440 : 160 : : 9 : 1 ft. Ans.
The celebrated Archimedes said he could move the Earth,
if he had a place at distance from it to stand upon, to manage his
machinery. % t
Now suppose the Earth to contain in round numbers
4,000,000,000,000,000,000,00000 lb. or 400000 Trillions of lbs*
and that Archimedes was suspended from the end of a lever
12,000,000.000,000,000,000,006,000 miles in length, and the
fulcrum, or centre of motion of the lever to be 6000 miles from
the Earths centre, how much must Archimedes weigh to balance
the Earth * Ans. 200 lb.
OF THE WHEEL AMD AXLE.
The proportion of the wheel and axle, (where the power is ap-
plied to the circumference of the wheel, and the weight to be
raised is suspended by a cord, which coyls about the axle as the
wheel turns round,) is as the diameter of the axle to the diatnc-
ARITHMETIC*. 43
terofthe wheel, so is the power applied Co the wheel) to the
weight suspended from the axle.
Suppose a windless is constructed in such a manner, that 14lb*
applied to the wheel will raise ?24lb. suspended from the axle,
which is 6 inches in diameter, what is the diameter of the
wheel ? Ans. 8 feet.
lb. in. lb. in.
Inversely 9 As 224 : 6 : : 14 : 96 = 8 feet.
Suppose the diameter of the wheel to be 8 feet,required the
diameter of the axle, so that 141b. suspended from the wheel,
may balance 2241b. on the axle.
lb. in lb. in.
Inversely, As 14 : 96 : t 224 : 6 diameter required.
Suppose the diameter of the wheel 96 in. and that of the axle
6 in; what weight suspended from tbe axle will balance 14ltu
upon the wheel ?
Inversely, As 96 : 14 : : 6 t 224 weight required*
OF LOGARITHMS.
THE operations of Multiplication and Division, when they
are to be often repeated, and the extracting of Roots, especially
if they be from the higher powers, become so tedious,, that it is
an object which has long employed the skill, and talents of the
roost profound mathematicians, to substitute in their place
jnorc expeditious, and easier methods of calculation.— -To effect
this, certain numbers have been so contrived, and adapted
to other numbers, thajt the addition and subtraction of the former,
fcave been made to perform the office of multiplication and divU
sion in the latter, with imcomparable facility find expedition.
The invention of Logarithms is by some ascribed to Baron
Napier. But the kind of Logarithms now in use, was invented
by Mr. Henry Brirggs, Professor of Geometry in Gresham Col*
lege, London.
LOGARITHMS (from logos, ratio and arithmos* number}
are the indices of the ratios of numbers to one another ; being a
series of numbers in arithmetical progression, corresponding to
others in geometrical progression.
-,, C 1 2 3 4 5 indices or Logarithms,
lftus I 1, 10, 100, 1000, 10000, 10000Q,
This is the most convenient series of numbers, to which most
of the modern tables of logarithms are calculated.
In which it is apparent that if any two indices, or Logarithms,
be added together, their sum will be the index, or .logarithm, of
that number, which is equal ta the product of the two terms, in
the geometrick progression, to which those indices, pr loga-
rithms belong.
Thus, the logarithms 2 and 3, being added together,make 5, cor*
responding to 100000, the product of 100, into 1000, and the
logarithms 1, and 4, being added together, make 5, the loga-*
rithm corresponding to 10000C, the product of 10 into 10000,
Whence it is evident that powers of the same root may be
multiplied) by -adding their exponents, or logarithms. In like
manner, if any one index, or logarithm, be subtracted from
another, the difference will be the logarithm of that number*
which is equal to the quotient of the two terms, to which those
logarithms belong. Thus j if from 5, (the logarithm of 100000)
ARITHMETICS. 45
be subtracted 2, (the logarithm of 100) the difference 3, will cor-
respond to 1000, the quotient of 100000 divided by 100.
Again; if from 5, (the logarithm 100000) be subtracted 3 9
(the logarithm 1000) the difference is 2, answering to 100, the
quotient arising from 100060 divided by 1000. Hence it is
manifest, that. ' ^
, A power may be divided by another power of the same root,
%y subtracting the logarithm of the divisor, from the logarithm
>f the dividend.
,J So also if the logarithm of any number be multiplied by the
index of its power, the product will be equal to the logarithm
of that power. Thus if 2, (the logarithm 100) be multiplied by
3, the product will be 6, equal to the logarithm of 1 000000, or
the 3d power of lot).
Again, if the logarithm of any number be divided by the in-
dex of its root, the quotient will be the logarithm of its root.
Thus, the index, or logarithm of 1000000, is 6, and if this
number be divided by 3, the quotient will be 2, which is the
logarithm of 100, or the cube root of 1 OOOpOO.
In the following series, to wit.
10* : 10 5 : 10* 10* 10° 10^ x 10-* JO-** 10-*
t JOQOO 1600 100 10 1 .10 100 , 1000 10000
T Whose Logarithms are
4 3 2' 1 — 1 , ' —2. —3 *-4 &c.
It will be seen, that the logarithms of all the numbers be-
tween 1 and 10, are greater than 0, but less than 1 ; since by the
series, it may be seep, that the logarithms of 1 and of 10, are O,
and 1.
; Thus the logarithm
of 2 is 0. 3010300
of 5 == 0. 6989700
, of 7= 0. 8450980
Each number therefore between I and 10, has for its index,
with a decimal annexed. [ '
Eor the same reason, if the given number be
V between ") the log. f\ and 2 1 + the decimal part
ISoind^OOO } **» be \ 2 and 3 L e - 2 + the decimal ***
1000 & 10000 J between t 3 and 4 3 + the decimal part
Thus the logarithm of the natural number
of 35 is \ 1. 5440680
of 175 is 2. 2430380
of 8795 is 3. 9442358
Whence' we derive this general truth. ^The index of the
logarithm, is always 1 less than the number of integral figures
in the natural' number^ whose logarithm is required ; or the in-
46 LOGARITHMICK
dex shows how many figures to the left, the natural number ex*
tends from the place of units.
Thus the logarithm of 35, is 1. 5440680. Here the number
of figures being two> the index or characteristic!*, ef the log-
arithm is 1.
The logarithm of 175 is 2 2430380. Here the nuiriber 17*
consists of three Jigures f the first of which on the left hand is
second from the place of units ; the index, or characteristick,
of the logarithm is therefore 2, and the logarithm 8795 is 3.
9442355. Which extending to three places counted from the
unite figure, must have 3 for the index of its logarithm. (
Integral numbers are said to form a geometrical series, in-
creasing from unity towards the left; but decimals are supposed
to form alike series, decreasing from unity towards the right?
the indices of whose logarithms are negative as has been shown
in the preceeding examples.
Whence it follows, that all numbers which consist of the same
Jtgures, whether integral, or fractional, or mixed, will have the
decimal parts of their logarithms the £ame y differing only in the
index, which will bemonr, or les8 y and fioritivey or negaiive r
according as the first .figure of the number is removed to the
right, or left, from the place of units.
Thus the logarithm of 735? is 3. 8668188 ; and the logarithm*
of ^y or t i v or tjfes &c. part of it, will be as follows.
Numbers.
Logarithms.
7359
3. 8668181
Ui 735.9
2. 8668188
73.59
1, 8668188
7.359
0. 8668188
.7359
— I. 8668188
.07359
— 2. 1668188
,007359
— 3. 8668188
Thus it appears that the,aegative index of a logarithm, s^ows
how far the first significant figure of the natural number, is re-
moved from the place of units on the right, in the same manner,
as a positive index shows how far the first figure of the natural
number, is removed from the place of units op the, left. But
wl?en the index of the logarithmic negative, it is oftenmore con-
venient to make it fiositive ; and this is done by adding 10 to
the negative index.
Thus, instead of — 1. 8668188") f 9. 8668188
of — 2. 8668188 IwriteX 8. 8668188
Of — 3. 8668188J 17.8668188
Bpcause -*- i + 10 = 9, and ~ 2 + 10 = 8, and ~m 3+ 10 /•
Although this in truth, makes the index 10 too great ; yet by a lit-
tle caution, it will pvoduce no ewour in the result ;— observing
ARITHMETICS At
always,; that when thesum, or product of the indices so increased,
exceeds 10, 10 must be rejected.
Thus the sunx of
— 2. 8668188 > . _ . C8. 8668188
_ 3. 8668188^ bCCOm<?S 2 r. 8668188
» _4. 7336376 6. 7336276
For— 4 +.10 = 6
And the product of — 3. 8668188"
Multiplied by 2 2
5. 7336376
For— .5 + 10 == 5
>or<
7. 8668189
2
CONSTRUCTION OF LOGARITHMS.
The usual method of computing the logarithms to any of the
natural numbers, 1. 2. 3. 4. 5. Sec. is, I believe, as follows.
RULE.f
1. Take any two numbers whose difference is unity, or 1,
and let the logarithm to the lesser number be known.
2. Divide the constant decimal, 868588964, &c. (or, 2-f — 2.
3025, Sec.) by the sum of the two numbers, and reserve the" quo-
tient ; divide the several quotients by the square of the sum of
the two numbers, and reserve the quotient ; divide this last
quotient also, by the square of their sum, and again reserve the
quotient ; and thus proceed, continually dividing the last quo-
tient, by the square of the sum Of the two numbers, aslong as divi»
sion can be made.
3. Then write these quotients in their order, under one anoth-
er, the 1st uppermost ; and divide them respectively by the
prime, or odd numbers, 1. 3. 5. 7. 9. 11. 13. 8c c. as long as di-
•The decimal parts of these logarithms are added as in sim-
ple numbers ; but when you come to the left hand figure of
each deeimal, there is -h 1 to carry to — 3, which equals — 2,
and this added to the — 2 above it,* gives — - 4 for the sum of
the indices.
t Yet there are many other ingenious methods of finding the
logarithms of numbers, (see Introduction to Dr. Huttons Tables,
and Baron Maseres Scriptores, Logarithmici, also, Kiel, on
Logarithms, Briggs Logarithms, Gardners, Taylors, Callets*
and Sherwins, Mathematical Tables.
i
48
LQGARITHMICX
vision can be made, that is, divide the first reserved quotient by
1, the second by 3, the third by 5, the fourth by 7, and so on.
4. Add all these last quotients together, and the sum will be
the logarithm Of the greater number divided by the less ; there-
fore to tl)is logarithm, add also the logarithm of the lesser num*
ber, and their sum will be the logarithm, to the greater, or pro-
posed number.
EXAMPLES.
Jlx ample 1st. Let it be required to compute the logarithm
of the number 2.
Here the given, or greater number is 2, and the next less
number is 1, (whose Logarithm, in every System, is always 0)
*lso the sum of 2 and 1 is 3. and its square 9 ; as follows.
3)
9)
9)
9)
«)
9)
a
9)
868588964
289529654
32169962
3574440
% 397160
44129
4903
545
61
1)
■o
•)
H)
13)
15)
289529654(
32169962(
3574440(
397160(
44129(
4903(
545(
61(
289529654
10723321
714888
56737
4903
446
42
4
Log. of 2—1.
Add. log. of 1
.301029995
.000000000
True Log. of 2. .301029995
Example 2rf.
Let it be required to compute the logarithm of 3
Here the given number is 3, and the next less is 2, whose
logarithm by the first example is .301029995, and the sum also,
of the 2 numbers 3 + 2=5 the square of which is 25, then the
operation is as follows.
*)
25)
25)
25)
25)
25)
25)
868588963
173717793
6948712
277948
11118
445
18
*>
5)
7)
9)
H)
173717793(
69487 12(
277948(
11118(
445(
18(
173717793
2316237
55590
1528
50
o
Log. of 3 4- 2
Add Log. qf 2'
.176091260
301029995
True log. of 3 required. 477121255*
• See « Mr. Hutton's practical rule for the constvtfctipa-.of
logarithms.''
Arithmetic*.
49
Iflien, because the sum of the Logarithms of numbers gives
the logarithm of their product, and the difference of the Toga-
arithmsgives the logarithm of the quotient of the numbers ; from
the above two logarithms, and the logarithm of 10, which is 1,
we may raise a great many logarithms, as in the following ex-
amples.
Example 3d. v Example 4th.
Because 2X2 = 4, therefore
To Logarithm of 2 .301029995 [
Add Log. of 2 .30 1029995 ; ;
*fhesumisL6gof4 .602059990 \\
Example 5th.
because 2X4= 8, therefore
To logarithm of 2 .301029995
Add log. of 4 .602059990
Because 2X3= 6, therefore
To logarithm of 2 .301029995
Add log. of 3 .477121255
Che sum is Log. of ,6 .778 1 5 1 250
Example 6th.
Because 10-^-2=5, therefore
From log. of 10 1.000000000
' r uke log. of 2 .30 1 029995
Give* the log of 8, 903089985 : Remains log. of 5 .69897000*
Example 7th.
Because 5 X 8 = 40, thereifore
To logarithm of 8 .903089985 ■
Add log, of 5 .698970005
Example 8th.
Because 8 X 40 = 320 therefore
To Logarithm 40 1 60205999*
Add log. of 8 .90308998ft
Logarithm of 320 2 50514997*
Logarithm of 40 1.6O205999O
And thus computing by this general Rule, the Logarithms t*
"the prime numbers, 2. 3. 7. 11. 13. 17. 19. 23. 29. 31. 37. 41.
43. 47. 8cc. and then by using composition and division, we may
easily find as many logarithms as we please^ or examine any log-*
arithm in the table*
ANOTHER METHOD op COMPUTING LOGARITHMS'
The construction of Logarithm according to the proceeding
Irules, given by the repeated extraction of Roots, is tedious \ the
simplest method yet known is the following.
* To make a Table of Logarithms*
\. Write for the logarithm of 1, a cypher for the index, and
as many cyphers for the decimal part of the logarithm as you
* See "Mr. Briggs differential method of constructing logf*
jMJthms."
*» LOGARITHMIC*
would wish the logarithms to be extended : for tire logarithm
of 10, write an unit, with the same number of cyphers ; for the
logarithm of 100, put 2, with as many cyphers ; for 100(5, put 3
for the index, with as many cyphers; for 10000, put 4 for the
index, &c.
2. Find the difference between some two logarithms above
1000, or rather 10000, that differ by unity ; multiply the two
numbers together, and that product by the constant decimal
43429448190325183896 &c.
3. Divide the last product by the Arithmetical mean between
the two numbers, and the quotient will be the logarithm of the
difference of the two numbers.
Thus ; Let it be required to find the difference between the
logarithm of 10000, and 10001. The product of these two
numbers is 100010000, which multiplied by 4343, &c. gives
43434343, which divided by 10000.5, the Arithmetical mean
between the two numbers, gives 4343. Now if to the logarithm of
1000 which is 4. 0000000,43 43 be added, we shall have 4.0000434,
the true logarithm of 1000 to 7 places.
Having thus found the difference of the logarithms of any
two numbers differing by unity, or 1, and consequently, some of
the logarithms, by dividing the difference found by the Arith-
metical mean of any two numbers differing by 1, we shall have
the difference of their logarithms.
Thus ; to find the difference between the logarithm of 274 t
ifcnd 275, divide 4343, the difference of the logarithm of 10000,
and 10001, by 274.5 the quotient will be 1582, the difference
required, which added to 2. 4377506 the logarithm of 274, gives
2. 4393327 for the true logarithm of 275.
5. Having by this Means found a few of the* prime logarithms,
the rest are made by Addition and Subtraction ; and having^
made the canon upward, above 1000 to 10000, by consequence
it is made for all infer io» numbers.
Directions for taking Logarithms nnd their numbers
from the Table.
To find the Logarithm of any number consisting of 4 figures*
Look for the number whose logarithm is required in the col-
umn of numbers, and against this number^ its logarithm will be
found. , ^
Thus ; the logarithm of 1234, is 3,0913151, so that* any num-
ber under 10000, may be easily found by inspection.
But if the number is greater than 10,000, but less than
10,000,000
*Cut off four figures on the left of the given number, and seek
the logarithm in the table ; add as many units to the index, as
AMTHMETICIU 61
there afce figures regaining on the right : subtract the logar-
ithm found, from the next following it in the table :— -then sis the
difference of numbers in the Canon, is to the tabular distance of
the logarithms answering to them, so are the remaining figures
of the given number to the logarithmick difference : which if
it be added to the logarithm before found, th* sum will be the
logarithm required. Thus, let the logarithm of the number
92375 be required. Cut off the four figures 9237, and to the
index of die logarithm corresponding to them, Add one unit, be-
cause one figure is cut off on the right.
Then from the logarithm of the next greater,
number. 9238, = 3.965578a
Subtract the logarithm of the
required number, 9237 = 39655309
* 10 471
♦Then as 10 : 471 t : 5 : 235
Now to the log. of 4.9655309
Add 235 the difference found.
And the sum is the log. 4.9655544 required.
Or more briefly ; find the logarithm of the first four figuru*
as before ; then multiply the common difference which stands
against it, by the remaining figures of the given number, from
the product cut off as many figures at the right hand, as you
multiplied by, and add the remainder to the logarithm before
found, fitting it with a proper index.
thus, 471 X 5 = 2355, cut off 5, and add 235.
To find, the Logarithm of a dicimal fraction.
The logarithm of a dicimal, is the same as that of a whole
number excepting the index.
Take out then, the logarithm of a whole number consisting
ef the same figures, observing to make the negative index
equal to the distancetof the first significant figure ofthe fraction,.
from the place of units.
The log. of 0.07643 is — 2. 8832639 f or 8. 8832639
of 0.00259 —3. 4132998*1 or 7. 4132998
of 0.0006278 —4. 7978213 t or 6. 7978213
To find the logarithm of a mixed dicimal number.
Find the logarithm, in the same manner as if all the figures
\
*If one figure is cut off, say as 10 is to the diff. of Log. if
two figures are cuteff, as 100 is to the diff. if three, as 1000 ?
Sec.
*3 fct>GARlTHMrc£
were integers ; and then prefix the index which belongs to th>
, Integral part.
Thus, the logarithm of 39.68 is 1. 5985717
Here the index is one, becouse 1 is the index of the logar
rithm of every number greater than K>, and less than 100.
lo find the logarithm of a vulgar fraction*
Subtract the logarithm of the denomenator from that of the,
numerator. The difference will be the logarithm of the frac-
tion. •
To find the logarithm of §£
Logarithm of 37 1. 56820 ip
logarithm of 94 1. 9731279
Diff. Log. of ££ — 1. 595073$
where the index 1 is negative.
To find in the Table the natural number to any logarithm.
This is to be done by the reverse method to the former, viz*
fcy searching for the proposed logarithm among those in the
Table, and taking out the corresponding number by inspection,
Id which the proper number of integers is to be pointed off, viz.
I more than the units of the affirmative index.
To find the number corresponding to a logarithm greater than
any in the Table. First, from the given logarithm subtract the
logarithm of 10, of 100, or 1000, of 10,000, till you have a loga-
rithm that will come within the compass of the table, find the
number correspondinjg to this and multiply it by 10, or 100, op
* 1000, or 10,000, the product is the number required. Suppose
/ for instance the number corresponding to the logarithm,
7. 7589875 be required * Y subtract the logarithm of the number
-* 10.000 which, is 4.0000000 from 7.7589875, there remains
3.7589875, the number corresponding to which is 5741, thia
multiplied by 10,000 gives the number answering to the given
logarithm.
METHOD OF CALCULATING BY LOGARITHM^
MULTIPLICATION,
RULE.
Take out the logarithms of the factors from the tabled
add them together and their sum will be the logarithm of the
product required. Then by means of the Table take out the
Natural number answering to the sum for the product sought*
'9LRITHMETICI5
&
EXAMPLES.
?. Multiply 45 by 27
Numbers. Logarithms.
45 1. 6632125
27 1. 4313638
product. 1215 3.0845763 Ans.
<S. Multiply.
lumbers. Logarithms.
23.14 1. 3643634
75.99 1 3807564
2. Multiply 709 by 13
Numbers. Logarithms.
709 2. 8506462
13 1. 1139434
9217
3. 964S896
Ans. 1758.4086
3. 2451198
.Multiply 3.7 by 3. 7
Numbers. Logarithms.
3.7 0- 5682017
3.7 0. 5682017
Ans. 13.69
1. 1364034
5. Multiply 3.586, 2.1046, 0.8372, 0.0294, all together.
Numbers. Logarithms.
3.586 0. 5546103
2.1046 0. 323 1 696 Here the 2 to be carried can-
0.8372 —,1. 9228292 eels the — . 2 and there re«
0.0294 — 2. 4683473 mains the —1 to be set down.
0.1857618 *— 1. 2689564.
0. What cost 87 pounds of grean tea; at g 2 12 cts.pr. Itt?
Numbers. Logarithms
212 0. 3263359
87 1. 9395193
Ans. g 184. 44 2. 2658552
7. What cost 160 bushels oats, at 50cts. pr. bushel ? Ans. g 89-
18. What cost 250 bushels of wheat, at g 1 60 pr. bushel ?
Ans. g400
*'. What cost 1260 lb. rice, at 5 cts. pr. lb. ? Ans. g 65'
DIVISION BY LOGARITHMS,
RULE.
From the logarithm of the dividend, subtract the logarithm of
the divisor, and the number answering to the remainder will be
the quotient required,
* Note. In every operation what is carried from the dicimai
part of the logarithm, to its index, is affirmative, and is therefore
V> be added to the index when \t is affirmative^ but subtracted
■wheji it is negative,
s*
ttOGARITHMICE
EXAMPLES.
li Divide 15811 by 163
Numbers. Logarithms.
Dividend 15811 , 4. 1989593
Divisor 163 2. 2121876
2. Divide 163 by B. IS
Numbers. Logarithm.
Dividend 163 2. 212187&
Divisor 8.18 0. 912753$
Ans.
97 I. 9867*717 | quotient 19.926 1. 299434$
3. Divide 100000 by 100
Number. Logarithms.
Dividend 100000 > 5. 0000000
Divisor 100 2. 0000000
Quotient 1000 3. 0000000
4. Divide 1000000 by l$0Q
Number. Logarithms.
Divid. 1000000 6 000000*
Divisor 1000 3. 0000000
quotient 1000. 3. OOOOOOfr
5. A ship took a prize worth 8 3960, it is required to divide it
equally among the sailors, who are 264 in number.
What did each man share it the prize ? Ans. 8 IS
If 125 lb. of sugar cost 2 26. 25cts. what is that pr. lb.?
Ans. 21 cts.
PROPORTION BY LOOARITHMSL
RULE.
If the proportion be direct, add the logarithms of the second
ftnd third terms, and from the sum subtract the logarithm of the*
first term. The remainder will be the logarithm. of the term
required*
If the proportion be inverse, add the logarithms of the first
And second terms, and from the sum subtract the logarithm of
the third. The remainder will be the logarithm to the required
term.
EXAMPLES.
Find a fourth proportional to 7964, 578, and 27960.
Numbers. Logarithms.
Second term 378 2.5774918
Third term 27960. 4.4465372
First term 7964
Fourth term f 327
7.0240290
3.9011313
$: 1228977
ARITHMETIC** S3
Find a fourth proportional to 768, 3frl, and 9780,.
Numbers. Logarithms.
Second term 381 2.5809250
Third term 978* 3.9903389
6,5712639
First term 768- 2.8853612
Fourth term 4852 3.6859027
♦ARITHMETICAL COMPLEMENT.
The difference between a given number, and 10, or 100, or
1©00, Sec. is called the Arithmetical Complement of that
number.
To obtain the Arithmetic At Complement of a number,
subtract the right hand significant figure from 10, and each of
the other figures from 9.
N. B. In taking the Arithmetical Complement of a logarithm,
if the index is negative, it must be added to 9 ; for adding a nega-
tive quantity, is the same as subtracting a positive one. The
difference between — 4 and + 8, is not 4, but 12.
THE ARITHMETICAL COMPLEMENT.
Of 5. 2473621 is 4. 7526379 '
of 1.9864362 is 8.0135638
of ©.6452310 is 9. 3547689
of-2. 7064923 isl 1.2935076
In the foihiDing firoft^rtion^ the calculation is made in botlp
ways,
1. If the profit on g 2625 employed in trade, is g 525, what is
the profit on g 7875 ?
* Note. The principal use of the Arithmetical Complement,
is, in working proportions by Logarithms ; for by this they may
be performed by merely adding' together the several terms of the
proportion.
*tf tOGARITHMICE;
By the common method.
Second term 525 2.7,201593
third term 7875 3.8962506
6.6164099
First term 3625 S.4191293
3y the Arith'l.Compilemerik ^
Second term 525 2.7201 59 »
Third term 7875 3.8962506
First term, a. c. 6.580870f
Fourth term g 1575 .3.1972806
Fourth term 1575 3.197280$
2. If g 567, gain g 81, what will be the gain on g 1701 ?
As 567 stock a,c. 7.2464169
Is to 81 profit 1.9084850
So is 1701 stock 3.2307043
To 243 profit 2.3856062
5. If the interest on g 450 for one year, is g 2?, what will be*
the interest on g 1150 for the same time. Ans. g 69 ?
4. When a pipe of wine costs g 252, what is the value of 174
gallons ? Ans. g 35 ?
5. Bought 721 yards broad cloth, at the rate of g 65 for every*
13 yards, what did the whole come to ? Ans. g 3605 ?
VULGAR FRACTIONS.
A vulgar fraction is any assignable part of a unit or integer,
expressed by two numbers, placed one above the other' with a
line drawn between them, as J one fourth, «£■ two thirds, Sec.
The number above the line is called the numerator, and that
below the line, the denominator.
A fraction is said to be in its lowest terms, when it is ex-
pressed by the least numbers possible, as £, when reduced to its-
lowest terms will be \ ; and T 9 T is equal to \ Sec.
CASE I.
To reduce fractions to their lowest term*.
rule. ,
Divide both the numerator and denominator, by any number
Which will divide them without a remainder, and the quotients
again in the same manner, till it appears that there is ho numbetf
greater than 1,' which will divide them again,.
EXAMPLES.
1. Reduce -JtI to its lowest terms. Ans. \
3. Reduce '-f^ to its lowest terms. Ans. -J*
3. Reduce f? to its lowest terms. Ans.. %*
ARITHMETIC** 5/
4. Reduce |ff to its lowest terms. Ana. {
5» Reduce -f J$£ to its lowest terms. Ans. J
Abbreviate $ffflfi as much as possible. Ans. 5SV1SV
CASE II.
To reduce the value of a fraction to the known parts of an
integer.
RULE.
Multiply the numerator by the common parts of the integer
and divide by the denominator.
EXAMPLES.
1. What is the value of f of a pound sterling ?
2
20 shillings in a pound.
Denominator 3)40(1 3s. 4d. Ans.
10
9
_ *
1
12 pence in a shilling.
3)12(4<k
12
2. What is the value of f pound sterling ?
Ans. £0 15s. O
3. Reduce { of an hundred weight to Us proper quantity.
Ans. 3qrs. 31b. loz. 12$dr,
4. Reduce f of lb. troy to its proper quantity.
Ans. 7oz. 4pwt.
5. Reduce $ of a mile to its proper quantity.
Ans. 6 fur. 16 po.
6% Reduce § of a month to its proper quantity.
Ans. 2 wks. 2d. 19 h. 12 m.
CASE III.
To reduce a fraction of one denomination to that of another,
but greater, retaining the same value.
RULE.
Reduce the given quantity to the lowest term mentioned, for
a numerator ; tfcen reduce the integral part to the same term,
for a denominator 5 which will form tfcte fraction required.
8
58 LOGARITHMICK
EXAMPLES.
1. Reduce 15s. 8d. 2qrs. to the fraction of a pound.
20 Integral part. 15.82 given sum.
12 12
240 188
4 4
»60 ueuominator. 754 JNum. jfj = f ±%£ .
2. What part of a pound sterling is 13s. 4d. ? Ans. §
3. What part of a hundred weight is 3 qrs. 14lb. ? Ans. f
4. What part of a yard is 2 qrs. 1 nail ? Ans. T * F
5. What part of a common year is 3 weeks, and 4 days ?
Ans. T %
6. What part of a mile is 6 fur. 26 po. 3 yds. 2ft.?
fur. fio. yd. ft. feet.
6 26 3 2 = 4400 Num.
a mile = 5280 Denom. fffj = £
7. What part of a hhd. of wine is 54 gallons ? Ans. 4
8. What part of a day is 16 h. 36 min. 55 T 5 j sec. Ans. T 9 ^
9. What part of a shilling is 4Jd. ? Ans. |
10. What part of an acre is 3 roods, and 20 rods ? Ans. £
1 1. What part of a pound troy is 10 oz. 11 pw*. 16 grs ?
Ans. if J
12. What part of a cord is 1 16-A- feet ? Ans. j-j
— -/> — —
DECIMAL FRACTIONS.
A Decimal Fraction is that whose denominator is an unit,
with a cyper, or cyphers annexed to it, thus T % T fo T7 5 ^ &c.
&c.
As the integer is always divided either into 10, 100> 1000 &c.
equal parts ; consequently the denominator will always be
either 10, 100, 1000, 10,000, &c. which being implied, need not
be expressed ; for the true value of a Decimal Fraction is prop-
erly expressed by writing the numerator, only with a point be-
fore it on the left.
f * 1 f 5
Thus, instead of < T 7 / 7 > write «< .75
I iW* J ' I -837 &c.
But if the denominator has not so many places as the denomi-
nator has cyphers, prefix so many cyphers on the left as will
make up the deficiency. * .
ARITHMET1CK. 59
Thus, for -{ u \ r > write 4 .C
L xoJor J l <
.05
.006
.0007 &c.
Decimals are reckoned from the left hand towards the right,
and the value of each figure is determined by its distance from
the place of units , if it be in the first place after unit,s (or sep-
erating point) it signifies tenth's ; if m the second, hundreth's
&c.dec reasing towards the right in a tend fold proportion, as in
the following
TABLE.
Crt (A
*•» «->
2 as,
G c: 2 A.J3 CCA
(0 3 3 CJ^ ft*? fi 3 3 w
C O O rt £ £ «* O © C
2HH0CC3 cgonHn:
SotfHKH^ HKHXoS
,765 4321 234567
* ^ 1 V ^-— J
Whole Numbers. Decimals.
Cyphers placed at the right hand of a decimal fraction do not
alter its value, since every significant figure continues to pos-
sess the same place ; so, 5, 50, 500, 5000, are all of the same
value, and each equal to / T or t
But cyphers placed at the left hand of decimals, decrease
their value in a ten fold proportion, by removing them farther
from the decimal point. Thus, .5 .05 .005 .0005 Sec. are
5 tenth, 5 hundredth, 5 thousandth paats, &c. respectively
ADDITION OF DECIMALS.
RULE.
1. Place the numbers, whether mixed, or pure decimals,,
under each other, according to their local value.
2. Find their sum as in whole numbers, and point off so many,
places for decimals, as are equal to the greatest number of de-
cimal parts in any of the given numbers.
Note. The point prefixed is called the Seperatrix, from,
its separating the Integral from the Decimal part.
40 liOQARITHMICK
EXAMPLES.
1. What is the sum of 276, 39.213, 72014.9, 417, 5032 and
2214.298?
/ , 276.
39.213
i 20 14.9
417.
5032.
2214.298
AhS. 79993.411
Etence we itiay observe that the Denominations of Federal
Monet, as determined by an Act of Congress Aug. 8, 1786, are
in a decimal ratio, and subject to one, and the same law of nota-
tion, and consequently of operation.
For since a dollar is the integer or unit , and a dime being
the tenth, and a cent the hundredth, and a mill the thousandth
part of a dollar, it is evident that any number of dollars, dimes,
cents, and mills, is simply the expression of dollars, and deci-
mal parts oi a dollar : Thus, 15 dolls. 8 dimes, 3 cents, 5 mills
is expressed in decimals. 15,835, or IS.-fJfo
2. What is the sum of the following expressions of money,
viz. R7530, g 16.201, g 3.0142, g 957.13, g 6.72819, g .03014 ?
Ans. g 8513.10353
3. Required the sum of .01 4, .9816, .32, .15914, .72913, and
.0047, Ans. 2.20857
4. What is the sum total of 27.148, 918.73, 14016, 294304,
.7138, and 221.7 ? Ans. 309488.2918
5. Required the sum of 512.984, 21.3918, 2700.42, 3.1£&
57.2, and 581.06, Ans. 3646.20888
SUBTRACTION OF DECIMALS.
RULE.
1 . Set the less number under the greater in the same manner
as in addition.
2. Then subtract as in whole numbers, and place the decimal
point in the remainder directly under the other points.
ARITHMETIC'S.
61
Dollars.
From 21.481
Take 4.90142
EXAMPLES
Rem
3. From
4. From
5. From
Feet
From 125.64
Take 95.5875$
Rem. 30.05244
Ans. 194.7925
16.57958
270.2 subtract 75.4075
2.73 subtract 1.9185
.9173 subtract .2138
6. From 407, subtract 91.713
7. From 800.135 subtract 16.37
8. What number added to 9.999999 will make 10?
Ans.
Ans.
Ans.
Ans.
o.ans
.7035
315.287
783.76$
Ans. One millionth part of an unit?
MULTIPLICATION OF DECIMALS.
RULE.
1. Set down the factors under each other, and multiply them
as in whole numbers.
2. And from the product) on the right, point as many figures
for decimals, as there are decimal places in both the factors.
But if there be not so many figures in the product as there
ought to be decimals, prefix the proper number of cyphers to*
supply th£ defect.
EXAMPLES.
1. Multiply 91.78
by .381
Multiply 520.3
by .417
product 34.968 1 8 product 2 1 6.965 1
3. Multiply .217 by .0431 Ans. .0093527
4. Multiply 5 1 .6 by 2 1 Ans. 1083.6
5. Multiply .051 by .0091 Ans. 0004641
6. What will 6.21 yards of cloth amount to, at 2 dollars 32
cents 5 mills pr. yard ? Ans. 2.325 X 6.21 =836.9954
7. What cost 27.13 lb. of green tea, at gl 12-J pr. lb. ?
Ans. 830.52125
8. What cost 53% lb. sugar^at 14-J cents pr. lb. ?
Ans. 87.79375
9. What will 12.125 acres amount to, at 865.25 pr. acre ?
Ans. 8791.15625
10. What is the value of .7584 ounces oi gold, at %\7.777 pr
oz. ? Ans. 813,482076'8
62 LOGARITHMICK
To multiply by 10, 100, 1000, &c. remove the separatrix, in
the multiplicand, so many places to the right, as the multiplier
has cyphers.
examples.
'7.853
78.53
The product of .7853<
Into 10 = 7.8530")
" 100= 78.5300* n
" 100a = 785.3000 f UV
J' 10000 =.7853.0000J
785.3
7853
DIVISION OF DECIMALS.
RULE.
Divide as in whole numbers ; and to know how many deci-
mals to point off in the quotient, observe the following rule.
1. There must be as many decimals in the dividend, as in
both the divisor and quotient ; therefore point off for decimals
in the quotient so many figures, as the decimal places in the di-
vidend exceed those in the divisor.
2. If the figures in the quotient are not so many as the rule
requires, supply the defect by prefixing cyphers.
3. But if the decimal places in the divisor exceed those in
dividend, add cyphers as decimals to the dividend, till the num-
ber of decimals in the dividend be equal to those in the divisor,
and the quotient will be integers till ail these decimals are used.
And, in case of a remainder, after all the figures of thj÷nd
are used, and more figures are wanted in the quotient, annex
cyphers to the remainder, to continue the division to any degree
of exactness. •
EXAMPLES.
1. Divide 5424.6056 by 43.6
43.6)3424-6056(78.546
3052
2580
8180
2616
2616
remains.
ARITHMETICS, /a
2. Divide 3877875 by.675 Ans. 5745000
3. Divide 7.25406 by 957 Ans. .00758
4. Divide 56 cts. by 1 doll. 12 cts. Ans .5
5. If 6.2 1 yards of cloth cost g 14.43825, what cost one yard ?
Ans. 22.325
6. What is the value of 1 lb. green tea, when 27.131b. cost
230.52125 Ans. 8 .12$
7. If an ounce of gold be worth g 17.777, what is the value of
.7584 of an ounce at the same rate ? Ans. £13.4820768
Note. When decimals, or whole numbers, are to be divided
by 10, 100,1000, 8cc. remove the separating point in the divi-
dend, so many places towards the left, as there are cyphers in
(he divisor.
EXAMPLES.
10 the quotient is 785.3
100 -' " 78.53
by V
7853 divided „, ^ 10()0 „ „ 7 g53
10000 " " .7833
j
REDUCTION OF DECIMALS,
CASE I
To reduce a Vulgar Fraction to its equivalent Decimal.
RULE.
Annex i^phers to the numerator, and divide by the denomi-
nator, the quotient will be the decimal required.
Note. So many places must be pointed off in the quotient, as
there were cyphers annexed to the numerator ; but if there be
not so many places of figures in the quotient, supply the defect
by prefixing cyphers on the left of the said quotient..
EXAMPLES.
i*. Reduce <§• to its equivalent decimal.
8)1.000
.125 Arcs.
2. Reduce £ to a decimal. Ans. .25
3. Reduce \ to a decimal. , Ans. .5
4. Reduce \ to a decimal. Ans. .2
5. Rednce \\ to a decimal. Ans. .85
6. What decimal is equivalent to ^V? Ans. .025
.7. Bring ^ T to a decimal. Ans. .09375
&. Find the decimal expression of ttit' Ans * ,co8
i
*4 LOGARITHMICK
CASE II.
To reduce number* of different denominations to their equiy*
alent decimal vplue*.
RULE.
Turing the given denominations to a vulgar fraction (by Case
IIJ. page 57) and reduce said fraction to its equivalent decimal
value.
OR
Rule 2. Write the several denominations above each other in
their order, placing the highest denomination at the bottom ;
tfonj beginning at the top, divide each denomination by its val-
ue in the next superiour denomination ; the last quotient will
be the decimal required.
EXAMPLES.
1? Reduce 15s. 9d. 3qrs. to the decimal of a pound.
15
12
189
4
2Q X 12 X 4 = S60)759.000000(.790625 Answer.
6720
8700
8&Q
6000
.«
5760
_
By RULE 2
2400
4
3,
1920
12
9.75
»»—
20
15.8125
4800
4800 Decimal .790625 required.
2. Reduce 12s. 6d. 3qrs. to the decimal of a pound.
Ans. .628125
3. Reduce 9s. to the decimal of a pound. Ans. .45
4. Reduce 19 s. 5d 2 qrs. to the decimal of a pound.
Ans, .9727916
5. Reduce 3s. 9d. to the decimal of a dollar. Ans. .625
6. Reduce 7 oz. 19 pwt. to the decimal of alb. Troy.
Ans. .6§25
7. Reduce 3 qrs. 2 1 lb. to the decimal of an c wt.
Ans. .93?[5
ARITHMETIC*.
SIMPLE INTEREST.
Ikterest is the premium allowed for the loan of money, relr
alive to which there are four particulars.
First, the Principal, or sum at interest.
Second, the Rats per cent, or interest of /*100, or dollars
for one year.
Fourth, the amount which is the sum of principal apd inter-
est, added together.
Interest is either Simple, dr Compound.
Simple Interest is that which arises from the principal
only.
RULE.
1 . Multiply the principal by the rate, and divide the product
by 100. The quotient is the answer for one year.
2. Multiply the interest for one year by the given number of
years, and the product is the answer for that time.
$. If there be parts of a year, as months, or days, work for the
months by the aliquot parts of a year, and for the days by simple
proportion,
Note. Solutions in Simple Interest exhibit the principle*
that the interest of £ 100, or £100, for 1 year, 2 years, 3 years,
Sec. correspond to a series,©/ numbets in arithmetical propor-
tion ; from whence will naturally arise the following Theorem*
that
If two ranks of numbers have the same ratio between every
pair of correspondents, then the numbers themselves, their cor-
respondent sums, and correspondent differences, will have th«
game common ratio.
Tbits
{
Principle. Int.
1 : 6
100 t fc
2 : 12
200 I 1*
3 : 19
or
300 : 1*
4 : 24
400 : 24
10 : 60 1000 : 60
In the first pair of ranks, the ratio between any two corre*>
pondents, is 6 ; therefore, taking any number from the first
rank,— suppose 2 ; then 2 : 12 : : 10 : 60; and 2 t 12 v z
£ : 48 ; or 2 : 12 : : 3 : 18. So also in the 2d. pair of,
rank's ;— 300 : 18 : : 400 : 24 \ and 300 :- 18 : : 1*9* : 6# >
or 300,: 18 : : 7(fe : 42 &cv
&
v £# ' JXXJARfTHMICK
EXAMPLES.
What is the interest pf £420 for 1 year, *\t £7 pr. cent, pir.
annum. Ans. r29 8*.
(1)
420 What is the interest of fltiS 16*. for
7 * 1 year, at 7 pr. cent ? Ans. £52 4*. 1 JoT.
.r- ( 2 )
29 | 40 745 16
20 7
«
8 | 00 . ^ 52 J 20 12
Ans. £29 8*. 20
4 J 12
12
1 J 44
4
1 | 76
3. What is the interest of £800, for 1 year, at 7pr. cent, w:
•nnum ? Ans. £56
< 4. Wiiat is the interest of £ 7 6, for 1 year, at 5 pr. cent ?
Ans. ^3 16*.
• 5. What is the interest of £2 11 5* for 1 year, at 7 pr. cent?
Ans. £14 15*. 9rf.
6. What is the interest of £472 1*. for 1 year, at 7 pr. cent,
pr. annum f^ Ans. £33 0*. lOJrf.
7. What is the interest of £270 10*. 6d. for 1 year, at 5 pr.
•ent, pr. annum? Ans. £13 10*, 6£rf.
8. What is the interest of g 542, for 1 year, at 7 pr. cent, pr.
annum ? * Ans. g 37 »94cts.
9. What is the interest of g 800, for otie year, at 6 pr. cent ?
, Ans. &48 00
10. What is the interest of g 875 35ceMs, for one year, at
6 pr. cent I Ans. g 52 52
11. What is the amount of a bond for g 387 50 cents, for one
year, at 6 pr. cent ? Ans. g 410 75cts.
Note 1. When the principal consists of dollars, multiply ky
the rate pr. cent ; the product will be the interest for 1 year, in
cents.
Note 2. Wfeen the amount is required, add the principal to
the interest.
ARITHMETICS. %f
CASE It.
1. If the interest required be for, years, months, end days, take
| the number of months, and set it under the place of tens, take
i part of the number of days and put it under the place of units
for a multiplier.
2. For the odd days, (if any) see what proportion they bear to
the week, and divide the prinoipal by this proportion, and then
proceed to multiply as in whole numbers ; the product will bo
the interest for the whole time, in dollars, cents, and mills.
EXAMPLES.
Required thfe interest of g 10 44cts. for 3 years 5 months and
10 days, at 6 pr. cent, pr. annum.
. y.m. tf.
f *= -ffil 10 44 3 5 10
, 2 06 12
i | 41
20+ 1 as 30 days.
10
6 I 40 [ 6
2,15,7,60 Ans. 2 dol. 1$ cts 36
■ „.',- 7 mills, T^tn. —
4 odd days = ^ of
. ./ ' • week.
2. What will 780 dols.. amount to, at 6 pr, cent, in 5 years 7
months and 12 days? Ans. 8 975 99cts.
3. What is the interest of g 824 15 cts. for 22 weeks, at 7 pr.
fcent ? Ans. g 24 40 cts. 7 m.
4. What is the interest of g 438 24 for 4 years 9 months and
14 days, at 7 pr. cent ? Ans. g 146 90cts. 7m.
CASE III.
When there is a fraction as %\ £, &c. in the rate pr. cent.
RULE.
Multiply the principal by the rate pr. tent, to the product add
£,£, &c. of said principal, and divide by 100 for the interest re-
guirerf.
3
48
3
48
62
K>88
64
<# BOGARITHMICK
EXAMPLES.
1. WhMis the interest of 8 428 for ©fid y*ar> *t*| pr. cent,
fr. annum?
51*28
6*
2568
t | 214
ior
for half.'
for one fourth.
g 28,8$ cts. Ans.
*. What is the interest of 216/* 5*. for one year, at 5f pr.*
t«nt ? , Ans* 11/, 17*. H<i
3. What is the interest of S 300 for one year, at 6 J pr. cent*
pr. annum ? Ans. g 18 75 cts.
CASE IV.
To find the interest of any sum of money, for any number of
fears and parts of a year.
1. Find the interest for I year, and multiply this by the giren
number of years.
2. If there be months and days, work for the months, by the
aliquot parts of a year, and for the days, by simple proportion.
EXAMPLES.
1. What is the interest of 64 dols. 58 cts. for 3 years 5 month?
%nd 10 days, at 5 pr. cent I Ans. g 11 12 cts, \^%m^
64 58 •
5
4 me» =t f
I tno. = J
10 days = 4
32290
3
Interest for one year in cents*
96870 for 3 years.
10763 for 4 months.
2690 for 1 month.
896 for 10 days.
Ans. g 11,12,19 = 1112 cts. or g 11, IS c.^l^^r
2. What is the interest of g 325 41 cts. for 3 years, and 4
months, at 5 pr. cent t Ans. g 54 23 cts. 5 m.
3. What will 3000/. amount to in 12 years and 10 months, at
Spr. cent? Ans. 5310/.
4. What will g 730 amount to at 6 pr. cent, in 5 years 7
months and 12 days J
Ans. g 975 99 cts.
AlUfriMEtlCKi
INSURANCE, COMMISSION, amd BROKERAGE*
Are allowances to Insurers, Factors, and Brokers, at a stipu-
lated rate pr. cent, as a premium for their services*
The same rules used in simple interest, apply to each of these
cases. '
1 . What is the commission oh £287 10 s. at 3j pr. cent ?
Ans. £10 1 s. S d.
2. A Broker sells goods for me to {^amount of £2575 \T s.
6 d. what is the brokerage at 4 s. pr. cwit ? Ans. £5 3 s. 0$.
3. What is the insurance of a house, trained at 8 1853, at TS
cts. pr. cent ? Ans. 8 13 89£ cfcu
DISCOUNT.
Discount is an allowance made for the payment of any sum
of money before it becomes due ; and is the difference between
that sum due some time hence, and its present worth.
The fireaent worth of any sum, or debt, due some time hence f
is such a sum, as, if put to interest, would in that time and at
that rate pr. cent, for which the discount is to be made, amount
to the sum, or debt then due.
What remains after the discount is deducted, is the firescnt
voorth.
RULE,
As the amount of 100/. or 100 dols. at the given rate and time :
is to the interest of 100 at the same rate and time, so is the giv-
#o sum to the discount.
Subtract the discount from the whole debt, and Ihe remainder
will be the present worth.
Or ; as the amount of 100, is to 100, so is the given sum to
the present worth.
PROOF.
Find the amount of the present worth for the time and rate
proposed} which must equal the given sum, or debt.
example.
What must be discounted for the ready payment of 100 dols;
due a year hence, at 6 pr. cent pr. annum ?
As 106 : 6 : : 100 : 5 66 Ans.
100,00 years sum.
5,66 discount.
$ 94,34 the present worth.
If LOGARITHMIC*
2. What sum in ready money, will discharge a debt of £9$Si
tfitc * ydar an<j 8 months hence, at 6 pr. cent? r
£ £ ££.*•*-
As 110: 100 : ; 925 : 840 18 2 Ans.
3. What is the present worth of 600 dols. due 4 years hence,
at5pr. cent? Ans. 8500
4. *What is the present worth of £*100, one quarter due in
3 months, and the remaining 3 quarters, in 5 months, discount
7 pr. cent ? Ans. £97 8 s. 10 d. +
5. What is the difference between the it terest of 81904* at
5 pr. cent pr. annum, £jf 8 years, and the discount of the same;
for the same time and rate ? Ans. 8 137 60 cts.
EQUATION OF PAYMENTS.
Is finding the equated time, to pay at once, several debts due
at different times, so that no loss shall be sustained by either
* party.
RULE.
Multiply each payment by its time, add the several products
together, and divide the sum by the whole debt ; the quotient
will be the answer.
PROOF.
The interest of the sum, payable at the' equated time, will
equal the interest of the several payments. '
examples.
1. A ewes B. 8380, to be paid as follows, viz. 100 in six
months, 120 in 7 months, and 160 in 10 months ; What is the
equated time for the payment of the whole debt ?
100 V 6= 600
120 X 7= 840
160 X 10= 1600
380 )3040( 8 months. Ans.
( . 3. The firm of B. & C. owe to the firm of B."& Co. the sum
ftf 8 300; payments as follows: 100 in 3 months, 100 in 4
months, and 100 in 6 months; required the equated time for
the payment of the whole debt ? Ans. 4 \ months.
Note. When Sundry sums are to be paid at different times,
find the rebate, or present worth of each payment separately
then add them into one sum.
ARITHMETIC*. %fi,
3. P. owes C. £420, which will be due 6 months hence, but P.
is willing to pay £60 «o«>, provided he can have the rest re-
main unpaid, a longer time than 6 months ; when must it be
jaid ? Ans 7 months^
FELLOWSHIP.
Fellowship, is a rule, by which Merchants, Sec. trading in
jpmpany with a joint stock, are enabled to ascertain each per-
sons particular share of the gain, or loss, in proportion to his
share in the joint*tock.
Fellowship is either single, or compound.
Single Fellowship.
Is when the several stocks in company are considered with-
out regard to time.
RULE.
As the whole stock is to each mans share in sibek, so is the
whole gain, or loss, to his share of the gain, or loss.
PROOF.
The sum of the several shares must equal the gain, or loss.
EXAMPLES.
1. A. B. and C. put in stock, and gain 800; A's stock was
1200, B. 4800, and C. 2000 : What was each mans share of the
gain ?
A. 1200
3. 4800
C. 2000
v****'
8000
* f 1200") fl2GY share of A.
As 800: i 4800 V: : 800: i 480 i — ^- B. An*V : '
^2000 J 4260 J — t— • C.
2. D. E. and F. trading, gained 120/. ; D's stock was 140K
E's S00,and F's 160 4 W,hat was each mans share of the gain.
« • • Ans. D's, 28/. E's, 60/. and FV32/1
3.' Four men* trading with a stock of g 2400, and gain in tf
years, twice as much, and g *60 more ; A's stock was 400, B'tr
740, C's 820 ; what was D's stock, and how much did each marf
gain. ■ _;
Note. By this rule,* also, a bankrupt estate, may be divided
among his creditors.
N
TSt iiOGARITHMlCK
TVs stock, £ 440 cts. grills."
A. gained R826 66 6
Ans. -^B. gained £1529 33 3
C. gained 81694T 66 6
D. gained £909 S3 3
COMPOUND FELLOWSHIP* x
Is when tbe respective stocks of several partners in company
are considered with time.
RULE.
Multiply each mans stock by its time, and add (he several
products together, then
As the sum of the products is to each particular product ; so
is the whole gain, or loss, to its share of the gain, or loss.
EXAMPLES.
1 . Three Merchants trade together. A's stock is , 1 20/, for 9
months, B's 100/. for 16 months, G's 100/. for 14 months, and
they gain 100/: What is. each, mans share.
A'sstock 120 X 9= 1080 : \.,- ;
B's — — 100 X 16 .= . 1600
G's ~- . lOO^X 14 =? 1400
4080 sum.
f 1080") f 26/. 9*. 4rf. i A. share.
As 4080 : -] 1600 I : : 100/."" < 39/. 4*. 3d. J B. share.
(^ 1400 J (_ 34/. 6*. 3d. £ C. share.
2. Three Merchants join in company. H. puts in 8 620 for 8
months, L. 950/. for 1 1 months, and M. 8 730 for 13ynonths, and
(hey gain 1 800 : What was each mans share ?
f A's share 358 55 4^
♦ . f Ans. 4 B's .755 42 1J£J .
'trr :'"■ IP's — -4j 686 02 f^V
i. A." began. trade, January i*.l 818* with a capital of g 1000,
S*id : on the first of March following, took in B. as a partner, with
a^api,ulof 8 1500 y three months after which, they admit C. a*
*t,ttiird partner, who brought into stock g 2800, and after trading,
together till the first of the next year, they find their gain to be>
$ 1776 50: How ought each oneishare in tl»e profit?
^.^ns. ^3 8457,46^11 B's g 571 83££J G's « 747$ff-
/
i
ARITHMETICS
i . _
ALLIGATION.
Teaches how to mix several simples of different qualities, so
that the composition shall be of a mean, or middle quality. It
consists of two parts, Alligation Medial, and Alligation Alternate.
ALLIGATION MEDIAL,
Is the method of finding the mean rate, or price of the com-
pound, by having the rates and quantities of the several simples
.given.
RULE.
Multiply each quantity by its rate ; then divide the sum of the
products by the sum of the quantities, the quotient will be the
rate of the compound required.
EXAMPLES.
1. Suppose 15 bushels of rye, at 64 cts. pr. bushel, 18 bushels
of corn, at 55 cts. pr. bushel, and 21 bushels of oats, at 28 cts.
pr. bushel, were mixed, what is the value of the composition pr.
bushel ?
bu. eta, g cts.
15 X 64 = 9 60
18 X 55 = 9 90
21 X 28 = 5 88
54 ~ 25 38 = 47 cts. Ans.
2. If 18 bushels of wheat, worth g 1 50 pr. bushel, be mixed
with 12 bushels of rye, at g 1 25 pr. bushel, what is a bushel of
this mixture worth I Ans. g l 40
3. Suppose a Wine Merchant mixes together 73± gallons of
wine, at g 2 16 pr. gallon, 5£ gallons, at g 2 pr. gallon, and 4£
gallons, at g 1 80 pr. gallon ; what will a gallon of this mixture
be worth, supposing he should accidentally spill a quart cf wa-
ter into it ? Ans gl 9S-J
4. A Goldsmith melted together 5 lb. of gold of 22 ci^ats
fine, 22 lb. of 21 carats fine, and 1. lb. of alloy ; what is the qual-
ity of the mass? Ans. 19 carats fine.
ALLIGATION ALTERNATE,
Is the method of finding what quantity of any number of in/-
gredients, whose rates are given, will compose a mixture of a
given rate : so that it is the reverse of Alligation Medial, and
may be proved by it.
RULE.
1. Write the several rates, or prices of the simples, in a,
column under each other, and the mean rate, c»r piiCv yf ihe
whole mixture, at the left hand.
iO
I
LOGARITHMICK
2. 'Connect, or link the price of each simples or ingredient,
which is less than the mean rate, or price of the whole mixture,
with one, or any number of those, which are greater than the
mean rate, and each greater rate, or price with one, or any num-
ber of the less.
3. Write the difference between the mean firice, (or mixture
rate) and that of each of the simples, opposite to the rates with
which they are connected.
4. Then if only one difference stand against any rate, it will
be the quantity belonging to that rate; but if there be several,
their sum will be the quantity.
EXAMPLES.
1. A Merchant would mix wine at 14s. 19s. 15s. and 22s. pr.
gallon, so that the mixture may be worth 18s the gallon ; what
quantity of each mu3t be taken ?
M«n !"!*;-)?"'«• or
— ^. , 0lis!£*"»^:g|:
1 22 — > 4 at 22s. 22— > 4
14 » 1 + 4
5 at 14s.
1 at 15s.
7 at 19s.
4 at 22s.
2. How much wine at 6s. pr. gallon, and at 4s. pr. gallon,
must be mixed together, that the, composition may be worth 5s.
pr. gallon ? Ans. 12 gallons of each.
3. A Merchant would mix several kinds of spirits together;
some at 7s. some at 10s. some at 5s. and some at 13s. pr. gallon ;
how much of each sort must the mixture contain, so that a gal-
lon of it shall be worth 9s. 2d ?
Ans. 4 gal. at 5s. 1 gal. at 7s. 2 gal. at 10s, and 4 gal. at 13s.
4. How much grain at 2s. 6d. 3s. 8d. 4s. and 4s. 8d. pr. bush,
must be mixed together, that the compound may be worth 3s.
lOd. pr. bushel ?
Ans. 12 at 2s. 6d. 12 at 3s. 8d. 18 at 4s. and 18 at 4s. 8d.
5. How much water at pr. gallon, may be mixed with li-
quors at Is. 7s. and 8s. pr. gallon, so that a gallon of the mix-
ture may be sold for 5s. pr. gallon ?
Ans. 5 of water, 5 at Is. 9 at 7s. and 9 at 8s.
* Note. By connecting the less rate with the greater, and
placing the differences between them, and the mean rate alter-
nately ; the quantities resulting, are such, that there must be
precisely as much gained by one quantity, as is lost by the
other; therefore the gain and loss upon the whole are equal,
and arc exactly the proposed rate. It is also obvious, that ques-
tions under this rule, admit of answers differing, ad infinetum ;
for having found one answer, we ma\ find as many more as we
please, by only multiplying, or dividing each of the quantities
found by 2, 3, 4, &c.
ARITHMETICS 75
TARE AND TRETT.
Tare and Trett are allowances made to the buyer, on some
particular commodities.
Tare is the weight of the barrel, box, bag, or whatever con-
tains the articles.
Trett is an allowance of 4 lb. in every 104 lb. for waste,
dust, &c.
Gross, is the weight of the goods together with the barrel,
box, bag, or whatever contains them.
When the tare is deducted from the gross, it leaves what is
called the suttle.
Neat, the weight of the goods, after all allowances are
made.
CASE I.
When the tare is a certain weight fir. box^ barrel^ Istc.
rule.
Multiply the number of boxes, or barrels, Sco. by the tare, and
subtract the product from the gross, the remainder is the neat
weight required.
EXAMPLES.
1. In 7fraits of raisins, each Weighing 5 cwt. 2 qr. 5lb. gross,
tare 231b. pr. frail; how much neat I
23X7 = 1 cwt. lqr.2llb.
S> 2, 5 What is the neat weight of 14 hogs-
T heads of tobacco ; each weighing
— — -r- 5 cwt 2 qr. 17 lb. gross, tare 100 pr.
38, 3, 7 gross, hogshead, Ans. 66 cwt. 2 qr. 14 lb.
1, 1, 21 tare.
S7, 1, 14 the answer.
CASE II.
When the tare is a certain weight fir. ciut.
RULE.
Divide the gross weight by the aliquot parts of a cwt., con-
tained in the tare, and subtract the quotient from the gross ; thet
remainder is the neat weight.
7£ LOGARITHMIC*
EXAMPLES.
1. Gross 173 cwt. 3 qr. 17 lb. tare 16lb. pr. cwt how mutch
peat ? *
cwt. gf. lb.
173 3 17 grow
"U lb. = | 21 2 26 2. What is the neat weight of 7 barrels
2 lb. = 4 3 11 of potash, each weighing 201 lb. gross,
— — — tare 101b. pr. cwt. I Ans. lS8lb.6©z.
24 3 9
149 8 Ans.
3. In 25 barrels of figs, each 2 cwt. 1 qr. gross, tare 16 pr*
Cwt. ; how much neat ? Ans. 48 cwt. 24ib*
CASE III.
When Trett i* allowed with Tare.
RULE.
Divide the suttle by 26, and the quotient will be the Trett,
which subtract from the suttle, the remainder is the net weight.
EXAMPLES.
1. In 9 cwt. 2 qr. 17 lb. gross, tare 37 lb. and trett as usual f-
how much neat ?
/*.
2. In 7 casks of primes, each
weighing 3 cwt. 1 qr. 5 lb. gross,
tara \7\ lb. pr. cwt. and trett as
usual ; how much neat ?
Ans. 18 cwt. 2 qr. 25 lb.
8 3 25 Answer.
CASE IV.
When Tare Trett and Cloffare allowed.
RULE.
Deduct the tare and trett as before, divide the suttle by 16$,
the quotient will be the cloff, which subtract from the suttle ;
the remainder is the neat weight.*
cwt.
9
gr. lb.
2 17
1 9
gross*
tare.
► J9
1 8
1 It
suttle.
trett.
ARITHMETICS 7t
• EXAMPLES.
1. What is the neat weight of a hhd. of tobacco, "weighing
15 cwt. 3. qr. 201b. gross, tare 7 lb. pr. cwt. and trett and cloff
as usual ?
cwt. or. lb.
2. In 19 chests of sugar, each
containing 13 cwt. 1 qr. 171b.
gross, tare 131b. pr. cwt. and
trett and cloff as usual ; — how
much neat ? and what is the
value 5^d. pr. lb. ?
Ans. 215 cwt. 171b. and value
£577 6S. Si
cwt. qr.
15 3
71b^ 3
26)14 3
2
lb.
20 gross.
27 ure.
21
18 trett
168)U 1
13 suttle
9 cloff
14 1
4 Ans
COMPOUND PROPORTION, OR DUBLE RULE
OF THREE,
Teaches to solve, at once, such questions, as require two, or
more statings in simple proportion, whether Direct, or Inverse.
In this rule, their are always five terms given, to find a sixth.
The three first terms of which, are a supposition, the two last a
demand.
RULE.
In stating the question, place the terms of supposition, so that
the principal cause of loss, gain, or action, possess the first
place ; that which signifies time, distance of place, in the second
place, and the remaining term in the third place. Place the
terms of demand under those of the same kind, in the supposi-
tion.
If the blank place, or term sought, fall under the third term,
the proportion is direct ; then multiply the first and second
Note. The following method of stating compound proportion
fa, by some, prefered,
1. Place that number, or term, which is of the same name, or
kind with the answer, in the third place.
2. Then take one term from the supposition, attd one from
the demand, both of the same name, or kind, and place thera
with the third term.
K 3. Then proceed in the^ same manner with the two remaining
terms.
78 LOGARITHMIC*
terms together for a divisor, and the other three for a dividend :
but if the blank fall under the first, or second term, the propor-
tion is Inverse; then multiply the third and fourth term togeth-
er for a divisor, and the other three for a dividend ; and the
quotient will be the answer.
4. Reduce the similar terms to the same denomination, if
necessary.
5. Multiply the terms in the second and third place together,
and divide their product by the product of those in the first
place j — the quotient will be the answer.
EXAMPLE*
If 7 men can build 36 rods of wall in 5 days, how many rods
can 20 men build in 14 days ?
. men 7 : 20 : : 36
days. 3 : 14
21 280
2 \) 10080(480
84
168
16-8
00
In compound proportion, therfe are always five numbers given
to find a sixth, which multiplied into the product of the two first,
shall be equal to the product of the other three. And it maybe
shown in compound, as in simple proportion, that the product of
the extremes, is equal to the product of the means ; thus
2X3: 12:: 3X4: 24; here the extremes are 2X3 and 24,
and the means are 12 and 3X4; now the product of 2 X 3 X 24
the extremes, is 144 ; and the product of 12 X 3 X 4 the means,
is also 144; whence it is evident that dividing either the pro-
duct of the three extremes, or mean» % by any two of the means
gives the other mean ; thus, 144 divided by the product of the
two means 3 X 4, gives 1 2 for the other mean ; and it is also
manifest, that by dividing the product of the three extremes, or
incantty by the product of any two of the extremes, the other
extreme is obtained ; thus dividing 144 the product of the
means, by 2 X 3 the product of two of the extremes, gives 24
for the other extreme : hence the propriety of the rule in mul-
tiplying the Sd. 4th. and 5th. terms, or means together, and di-
viding by the product of the two first terms, or extremes, ta
obtain the other extreme.
ARITHMETIC*. $»
EXAMPLES. %
If 7 men can build 36 rods of wall in 3 days, how many rods
can 20 men build in 14 days ?
7:3:; 36 terms of supposition.
20 14 terms of demand.
56
If 150/. in 12 months, gain 9/. in what
time will 450/. gain 54/. ?
150 : 12 : : 9
450 : : 54 Ans. 2 years.
7X3 = 21)10080(480 rods. Ans
BY LOGARITHMS.
In compound, as in simple proportion, the term required may
he found by logarithms, if we substitute addition for multiplica-
tion, and subtraction for division.
RULE.
Add together the logarithms of those terms which in com-
mon arithmetick are to be multiplied together for a dividend,
and from the sum,, subtract the sum of the logarithms answer-
ing to these terms, which in common arithmetick, are multipli-
ed for a divisor ; and the remainder will be the logarithm of the
answer. Or more fully ; find the arithmetical complement of
the logarithms to be subtracted, and then add all the terms to-
gether ; the sum will be the logarithm of the answer.
examples.
2. If 4 men in 12 days, mow 48 acres ; how many acres can
8 men mow, in 1 6 days . ?
Two first terms j 1 2
Third term 8
Fourth and fifth terms < lfi
* Term required 128 Ans. 2. 1072100
a. c.
9.
3979400
a. c.
8.
9208188
0.
9030900
1.
6812412
1.
2041200
* It must be observed, that each arithmetical compliment in-
creases the index of the logarithm by 10 ; as often therefore,
as the a. c. is used, the index of the sum of the logarithms,
must be made less by so many tens.
& LOGAWTHMICK
1. If 10 bushels of oats, be sufficient for 18 borses 20 days*
.How many bushels will serve 60 horses 36 days ?
T60 log. 1. 7781513
Three last terms i 36 l. 5563025
(.10 1. 0000000
Sum of Log. 4. 3344538 *
L*g. to be subtracted JJJ log
1. 2552725
1. 3010300
2. 5563025
Sum of the Log's, three last terms 4. 3344539
do. of the two first 2. 5563025
Term required 60 =* 1 . 778 15)3
3. What principal will gain £*2&2 10s. in 7 years, &\£5 pr«-
cent I Ans. £750
4. If the interest of 365 dollars for 3 years and 9 months, be
g 82 13 cts. what will be the interest of g 8 940 for two years
and 6 months ? Ans. 1 340
5. If/100 in 12 months, gain £6 interest; how much will
£"75 gain in 9 months ? Ans. £ 3 7s. 6d.
6. If jf 16 18s. be the wages of 16 men, for 8 days ; what sum
will 32 men earn in 24 days ? Ans. £ 01 8s.
7. If £ 100 in 12 months, gain £7 interest ; what principal
will gain £3 18s. 9d. in 9 months i Ans. £ 75
8. If 2001b. be carried 40 miles, for 40 cents; how far may
-202001b. be carried for g 60 60 cts. Ans. 60 miles.
COMPOUND INTEREST.
In calculating compound interest, the amount for the first year*
Is made the principal for the second year ; and the amount for
the second, the principal for the third, Sec.
As the Logarithmic ka I method of computing compound inter*
est, is by mutch the most cxpediiious, it is thought unnecessary
to subjoin the old one in this place. Therefore,
To calculate Compound Interest by Logarithms.
RULE.
1. Find the amount of 1 dollar for 1 year; multiply its loga-
rithm by the number qf years, and to the product add the loga*
abjthmetick; jj, :
rithm of th£ principal. The sum will he tfye ^arkhm of tic
Amount for the given time.
2. From the ampunt subtract the principal, and the remainder
will be the interest.
EXAMPLES.
1. What is the amount of 20 dollars, at 6 pr. cefit compound
interest, for 100 years ?
Amount of 1 dollar for 1 year = 81 06 log. 0.0255059
Multiply by the time \<K)
2.5305900
Add log. of S 20, given principal 1.3010300
Amount required 8 6786 3.8316200
2. What is the amount of 425 dollars, at S pr. cent compound
interest, for 4 years ?
Amount of 1 dollar for 1 year == g 1 05 log. 0.021 1893
Multiply by the time 4
0.0847572
Add log. of 425 given principal 2.6283889
Amount 8516 59 3.7131461
Note 1 . If the the interest becomes due semianually^ or guar*
ttrly ; find the amouaf of one dollar, for the half-year or quar-
ter, and multiply the logarithm, by the number of half-years or
quarters in the given time.
Note 2. As Simple Interest is performed by a rank of numbers,
arithmetically proportional, so it may be shown, that Compound
Interest is performed by a rank of numbers geometrically pro-
portional.
And it is a principle in Mathematicks, that, if three num-
bers be in geometrical proportion, the product of the two ex-
tremes, is equal to the square of the mean. (See Euclid's Eli->
ments, 20th prop. 7th book.) And on the contrary, if the rec-
tangle contained by the extremes of any three numbers, be equal
to the square of the mean, then those three numbers are in.
geometrical proportion.
Now if 3 dollars be the compound interest of 8 100 for \ year,
or 6 months, then these three numbers 100,103, 106, should be in
geometrical proportion ; but it may be proved by the aforesaid
proposition, they are not; for the reetangle of 100 into 106 is
but 10600, and the square of the mean 103, is 10609, which is
greater than the product of the two extremes. But the square
root of 10600 will be found to be 102,956 : so that the true pro-
portional interest of g 100, for i year, is but g 2 95 cts. 6 m. t
11
A» LOGARITHMIC*:
3. What i* tte anuMintof 1000 dollars, a* 6 pr. cent
pound interest) for 10 yean* ? 4ns. $1790
4. Required the amount xrfiOD cblla», at 6 pr. cent compound
interest, for 3 years? Ans. 8 119 10
5. What will 1000 dollars amount to at 7 pr. cent, compound
interest, in 4 year? ? Aas. 8 1310 80
6. What is the compound interest of 876 dollars 90 cts. at 6
j>r. cent pr. annum, for 3 yeai? and 6 months? Ans. $ 198 83-f-
7. What will 100 dollars amount to in 3 years, at 6 pr. cent
compound interest} allowing that it becomes due semiannually £
Ans. 2 127 054-
8. What is the amount of 400 dollars, at 5 pr. cent com;
pound interest, for 1 year, payable quarterly ? Ans. 420 37
9. What is the amount of 1 cent, at 4 pr. cent compound in*
fewest, in 500 years ?
Amount of 1 dollar for 1 year== 8 1 06 log. 0,0253059
Multiply by the time 500
I I I II MMMM
12.6529500
Add log. of principal 8.0 01 — 2.0000000
Amount 8 44,973,000,000 IQ;6329$6b
INVOLUTION.
A Power is a number produced by multiplying &y given
number continually by ktelf a certain number of times.
The number denoting the power, is called the Index* jonr
Exponent of that power*
To rai&c a ghten number, woe hattt the foltevtins
RULE.
Multiply the given number, or first power, continually by it?
self, tin the number of multiplications be 1 less than the index
of the power to be found, and the last product will be the power
required.
Note. Powers are commonly denoted by writing their indices
above the first power; as follows.
2X2=4, the 2d power, or square of 2, or 2*.
£ X 2 X 2 a 8, the 3d power, or cube of 2, or 2',
t X 2 X 2 X 2 = 16, or hiquadfete of h <« 2 ** * c -
ARITHMETICS u
XXAtfMSS*
h Let it be required to raise 45 to its cube, or third power ?
45
10125
8100
Ans. the 3d power, 91125 or cube of 45
2. What is the square of 3758 ? Ans. 14122564
3., What is the cub* of 327 ? Ans. 3-4965783
4. What is the bftfuadfltte, or fourth power qf 376 ?
Ans. 19987173376:
5. What is the fifth power of -029 ? Ans. -000000020*1 1 149
6. What is the sixth power of 48 ? Ans. 12330&9O464
7. Required the seventh power of 7 ? Ans* 823543
. EVOLUTION-
Is that, by which we extract the roots of number*; or find a
radical quantity, which multiplied into itself a certain number of
times will produce the given power.
1*0 EXTRACT THE SQUARM ROOT.
RULE*
1. Having distinguished the given number in periods of two
figures each, beginning at the place of units, find the greatest
square number in the first, or left hand period, place the root of
it at the right hand of the given number, (in the manner oi a
quotient figure ia division,) for the first figure of the root, the
square of which subtract from the first period, and to the re-
mainder bring down the next period for a dividend.
2. Pkcte the double of the root, already found, on the left of
the dividend for a, divisor. .
Note. Roots are sometimes denoted by writing </ before the
power, with the index, of the root against it. Thus the third
r,oot of 80 is ^ i B0, an4 the second root of 80 is V 80 > the index
T y though omitted, is always to be understood, when the radical
sifep is written without a numeral index,
**
LOGARITHMICK
3. Consider what figure must be annexed to the divisor, so
that if the result be multiplied by it, the product may be equal
to? or next teas than the dividend, and it will be the second fig-
ure of the root
4. Find a divisor as before, by doubling the figures already
in the root ; and from these find the next figure of the root, as
in the last article ; and so on through all the periods to the last.
/ EXAMPLES.
1. What is the square root 2.
of 14122564? of
14* 12*25*64 | 3 the root'
9
What is the square root
5499025 ?
5.49*90-25 I 2345 root.
4
67 I 512
469
43 I 149
129
745 I 4325
3725
7508 I 60064
60064
464 I 2090
1856
4685 J 23425
23425
remains.
3; What is the square root of 10342656 ?
4. What is the square root of 43264 ?
5. What is the square root of 451584 ? *
6. What is the square root of 2985984 ?
T. What is the square root of
$. What is the square root of
998001
remains.
Ans. 3216
Ans. 208
Ans. 672
Ans. 1728
Ans. 999
9*4,5192360241 ?
Ans. 31,05671
9. What is the square root of 103089* 198,4001 ?
Ans. 32107,51
1 0. * What is the square root of 1 60 ? Ans. 1 2,649 1 1 4-
1 1 . What is the square root of 2 ? Ans. 1 ,4 1 42 1 35 6237 +•
12. What is the square root of 10 ? Ans. 3.162277 &c.
* Note. When the given number is a surd ; that is, when its
root cannot be found exactly, without a remainder, the evolu-
tion may be carried on, until we obtain a root, sufficiently near
the truth, by annexing cyphers to the remainder, and proceed-
ing as in rational numbers. In the 10th example ; although
12.6491 1, is not the exact root of 160, yet if it be multiplied by
itself, the product will be 159,9999837921, which is not two
parts, of which 10000 taskc an unit, wide jof thetrfltlu,
ARITHMETIC*. 8*
TO EXTRACT THE CUBE ROOT.
RULE.
1. Having distinguished the given nuirfber into period* of
three figures, find the nearest less cube' in the first period, set
its root in the quotient, and subtract the said cube from the
first period; to the remainder bring down the second period*
and call this the Resolvend.
2. To three times the square of the root, just found, add three
times the root itself, setting this one place farther to the right
than the former, and call this sum the Divisor. Then divide
the resolvend, excepting the right hand figure, by the divisor,
for the next figure of the root, which annex to the former ;
calling this last figure e> and the part of the root before found,
call a.
3. Add together these three products, namely, three times
the square of a multiplied by e, three times a multiplied by the
square of ?, and the cube of e\ setting each of them one place
farther towards the right than the former, and, call the sum the
Subtrahend x which must not exceed the resolvend ; if it
does, then make the last figure e less, and repeat the operation
for finding the subtrahend.
4. Subtract the subtrahend from the resolvend, and to the
remainder bring down the next period of the given number for
anew resolvend ; to which form a new divisor from the whole
none root found ; and thence another figure of the root, as before
and thus continue till the wh<#e is finfehed.
•6 LOGARITHMICK
EXAMPLES.
H Required the cube root of 4360368248*7 >
3X7* = t& | 436-036M4-2ST \ 7583 ttfe Mot:
3X7 = 21 J 343
l%\. 9msor. 1511
93036 1st. resofvend.
78875 subtract subtrahend*
r 3X7*X5 735
Add J3X7X5»=525
I 4! 6 1824 2d. resolvend.
1 36445 f 2 sub. 2d. subtrahend.
* 5* = 125
1st. 78875
517312287 3d. resolvend.
517312287 sub. 3d. sub'end.
3X75 * » 1787& remains.
3X75 gs 225
2d. Diviso* IT8975
f 3X75*X8 ^ 135000
Add« 5X75X8* =*= 14400
I 8* =- 512
2d. Subtrahend
3Xr58* =*
3X75*
13644512
1725692
2274
Gd. Divisor.
f 3X7582X3 =
Add J 3X758X32 «=
1 3' «
17239194
5171076
20466
27
3d. Resolvcnd
S1731228r
The laborious operation of extracting the roots of higher
pewers, is often so tedious, as to render it highly irksome and
'forbidding to learners. But, as in Logarithms, addition is
made to perform the office of multiplication, and subtraction
that of division ; the labour of evolving roots is not only short-
ened to a degree surpassing credence, but the whole h per-
formed with incomparable facility and expedition.
ARITHMETICS.
87
EVQUJTJON BY LOGARITHMS.
Evolution is the opposite of involution. And it was shown
in the introduction of logarithms, that quantities are involved, by
multiplying their indices^ or logarithms. For the same reason,
therefore, the roots of quantities may be, extracted, by dividing
their indices, qr logarithms.
To extract the root of any number by logarithm^.
We have therefore, this general
RULE.
Divide the logarithm of the given quantity, by the number
expressing the root to be found.
examples.
1. Required to find the cube root of the same number
436036824287, by logarithms ?
Numbers. Logarithms.
Power 436026824297 3 j 11. 6395233
Root 7583 3. 87984 ii
2. What is the squre root of 92613
Numb.
Power 9201
Root 21
3/ What is the square root of 9801 ?
Numb,
Power 98#l
Root 99
4. What is the square rpot of 365 ?
Numb.
Power 365
Root 19.10498*
5. Required the cube root of 12345 I
Numb.
Power 12345
Root 23.11162
«. What is the 10th root of 2 ?
Numb.
Power 2
Rbbt 1.000121
Log.
3 J 3. 9669579
1, 3222193
Log.
2 J 3. 9913704
1. 9956352
Log.
2 | 2. ,5622929
1. 2811465
Log.
3 I 4. 0914911
1. 3638304
Log.
10 | 0. 3010300
0. OSOlOof),
18
LOGARITHMIC*
7, Required the 10th root of 6948 ?
Numb.
Power 6948
Root 2.422
8. What is loeth root of 983 ?
Numb.
Power 983
Root 1.071
10 [ 3. 6418598
0- 3841859
Log.
£00 | 2. 9925535
0. 0992553
9. Required the 365th root of 1.045 I
Numb. Log.
Power 1.045 365 | 0. 0191163
Root 1.0Q121 0. 0000524
♦10. Required to find the 10000th root of 49680000 ?
Numb. Log.
Power 49680000 10000 | 7< 6961816
Root 1.0017899 0. 0006961
The Logarithms of Povters given, to find their roots.
X. Required the square root of 6561
Powers. Logarithms* Roots.
6561 3. 8169700 Ans. 81
2. Of
4096
3. 6123599
Ans. 64
3. Of
15625
4. 1938200
Ans. 125
4. Of
46656
4. 6689076
Ans. 216
5. Of
M7649
5. 0705882
Ans. 34S
6. Of
262144
5. 4185400
Ans. 512
7. Of
531441
5. 7254550
Ans. 729
8. Of
1679616
6. 2048674
Ans. 1266
9> Of
5764801
6. 7607844
Ans. 240 1
to. Of
43046721
7. 6339400
Ans. 6561
Required the cube root
Powers*
1. Of 1728
of the following \
Logarithms.
3. 2375437
numbers.
Roots.
Ans. 12
2. Of
8000
3. 9030900
Ans. 20
3. Of
15625
4. 1938200
Ans. 25
4, Of
19683
4. 2940914
Ans. 27
5. Of
10077696
7. 0033614
Ans. 216
6. Of
244140625
8. 3876400
Ans. 625
7* Of
68719476736
10. 8370797
Ans. 4096
♦Note. We have here an instaqpe of the great rapidity with
which arithmetical operations are performed by the use of Lqx>-
ARITmfS.
ARVTKMETJCR. W
«. Ui yyvo'00029999 11. 9998698 Ans* 9999
9. Of 205884571094649 14. 3136375 Ans. 59049
20. What is the fourth root pf 19987173376 ?
Log. = 10, 3007512 sb Ans. 376 the root.
11. What is the fifth rpot of 507682821 106715625 ?
Log. « 17. 48810?0 « Ans. 3145
12. What is the sixth root of 43572838 1009267809889764416?
Log. = 26. 6392164-7-6 = 4. 4398694 «= ${7534 the Aos.
1 3. What is the seventh root of
34487717467307513182492153794673?
Log. = 31. 5376642 --7 = 4. 5053806 = 32017 Ans.
14. What is the eighth root of
1121016231320476236246497942460481?
Log. = 33. 0496120—8 = 4. 1312015 = 13527 Ans.
Nothing can be more easy, than to extract the roots of powers,
to which the. logarithms are givet, or may easily be fou*d. But a
difficulty may sometimes arise, in the learner'sjfotfiftj' the exact
logarithm to a proposed number, that is much greater than any
in the Tabus; yet a very superficial attention) to the nature of
logarithms, will readily suggest a solution of this seeming diffi-
culty ; for, as adding the logarithms of several numbers, is
equivolent to multiplying by the same numbers, and subtracting
the logarithm of numbers, the same, as dividing by those num-
bers; therefore,
To find the logarithm to a proposed J&umber greater than any
in the Table.
RULE.
Resolve the given quantity into such factors, as will consti-
tute it within the limits of the table, add together the logarithms
of these factors, and the sum will be the logarithm to the pro-
posed number.
3P*AMPJU*S.
I. Required the logarithm of the natural nunaber, 11042f
Here it is evident, that dividing the. given quantity by two, will
constitute it within the limits of the Table ; as follows,
Factors,
3)11042(5521 Log. 3. 7420177
2 Log. 0. SO 10300
Given number. 11042 required Log. 4, -0430477
12
9ft
LOGARITHMIC*
2. Find the logarithm of 15378*
Factors.
3)15378(5126
Ltfg. 3. 7097785?
Log. 0. 4771213
Given number I5S78 required L<%. 4. 1868999
3. Required to'fl^d the logarithm of 17304
Factors.
4)17304(4326
4
Log.
Log.
, 3. 6360805
0. 6020600
Given number.
17304
required Log.
4. 2381405
4.
5.
What is the logarithm
Of 19505 ?
Of 25596 ? *
Ans.
Ans.
Logarithms.
4. 2923068
4. 4081722
6.
Of 39126?
Ans.
4. 5924655
7.
Of 57320 ?
Ans.
4. 7583062
&
Of 71464 ?
Ans.
4. 8540873
9.
Of 89897 ?
Ans.
4. 9513229
40.
Of 119844?
Ans.
5. 0786162
11.
Of 217975 ?
Ans.
5. 3384067
12.
Of 3089725?
Ans.
6. 4899199
CASE II.
When there are cyphers on the right hand of the given num-
ber.
RULE.
Find a logarithm to the significant figures, as before, and in-
crease the index by as many units, as there are cyphers on the
right of the givep number.
EXAMPLES.
1. What is the logarithm of 57640 ?
The logarithm of 5764 is 3. 7507240 ; pnd increasing the
index 3, by 1 j we shall have 4.
57640
2. Required the logarithm
Of 586400
3. Of 6495000
4. Of 72970000
5- Of 910100000
6. Of 44.973.00^000
7607240 for the logarithm of
Ans.
Ans.
Ans.
Ans,
Ans.
5.
6.
7.
8.
19-
7681940
8125792
8632634
959089 1
6529:5 GO
AftlTHMEtlpK. 91
JPlRjtCTICAL QUESTIONS W EVOLUTION AMI*
INVOLUTION.
BROBLEM I.
To find & mean proportional between two numbers.
RULE.
Add together the logarithms of the given numbers, divide
the sum by 2 ; the quotient will be the logarithm of the mean-
proportional required.
1. Required the mean proportional between 45 and 180..
1. 6532125
Thus \ 180= 2. 2552725
f 45 =
2)3. 9084850
Mean proportional required 90 e= 1. 9542425
2. Required a mean proportional to the numbers 64 and 256
Ans. 128
PROBLEM II.
Any number of soldiers being given, to place them in a square
Battalia of men.
RULE.
Divide the logarithm of the given number by 2 ; the quotient
will be the logarithm of the answer.
3. Let 9216 men be ordered to form a square battalia ; how-
many must stand in rank and file ?
9216 2)3. 9645425 Log.
Ans. 96 1. 9822712
4. How many must stand in rank and file, so that 5625 men
shall compose a square ? Ans. 75
5. Let 8450 men be so formed, as that the number in rank
may be double the number in file.
8^50^-2= 4225 = log. 3. 6258267 ~ 2 = 1. 8129J33 = 6s
2)8450
4225 log. 2)3. 6258267
Ans. 65 in file * 1. 8129133
Multiply 2
And 130 in rank.
Note. When the question requires double, trible, or quadru-
ple, the number of men to stand in rank, as in file ; divide the
logarithm of t, -f, £ &c. of the given number by 2, the quotient
wll be the number in file, which double, triple, quadrupfo
Sec. apd the product will be the. number in rank.
i
9& LOGARITHMIGK
6. Required to set out 27648 fruit trees, so that the number
in length, $hall bfe to the number in breadth as 3 is to 1 ; how,,
must they be placed ?
Ans. 288 in length, and 96 in breadth.
PROBLEM III.
Any two sides of a rightangled triangle being given, to find
the other side. *
Case 1. When the base and perpendicular are given, to find
the hypothenuse.
RULE,
Add the squares of the two legs together, and extract the
square root of the sum.
7. A triangular piece of ground measures 30 rods on one side,,
and 40 rods on another ; required the length of the remaining,
or longest side.
30 X 30 = $00
40 X 40 ex 1600
3500(50 hypothenuse, or
25 longest side
oo -
8. Required the length of a brace in a building, so that the
lower end of it shall be 8 feet, and the upper end, 6 feet from
the right angle. Ans. 10 feet.
9. Suppose the lower end of the brace to rest in a post 3 feet,
and the upper end framed into a plate 2 feet 3 inches from the
right angle : required its length. Ans. 3 feet 9 inches.
10. What will be the length of a brace, when it is required
that the distances from the right angle to either end, should be
2 feet 6 inches, and 3 feet 4 inches ? Ans. 4 feet 2 inches.
1 1 . Two stages start from the same- place ; one goes directly
south at t^e rate of 9 miles an hour, for 3 hours, the oiher due
west, for 4 hours, at the same rate ; in what time would they
now meet, were their course turned directly towards each other,
continuing at the same rate per hour ? Ans. 2-J- hours.
12. Required the length of a scaling ladder, to reach the top
of a wall, whose height is 28 feet, the breadth of the ditch be-
fore it being 45 feet ? Ans. 53 feet.
♦Note. The square of the hypothenuse, or the longest side
of a rightangled triangle (by 47th. Theorem B. 1. Euc.) is
equal to the sum of the squares of the other two sides ; and
consequently the the, difference of the squares of the hypothe-
nuse and either of the other sides is the square of the remain-
ing side.
ARITHMETICS. §*
PROBLEM IV.
CASE 2.
The hypothenuse and one leg. being given, to find the other
log. /
RULE.
Subtract the square of the given leg. from the square of the
hypothenuse, and extract the square root of the difference.
13. What is the perpendicular of a right angle triangle, whose
base is 56 feet, and hypothenuse 65 ? Ans. 33 feet.
14. What is the base of a right angled triangle, the hypothe-
nuse being 159 feet, and the perpendicular 84 feet ?
Ans. 135 feet.
15. A line of 65 yards will reach from the top of a precipice,
standing close by the side of a brook, to the opposite bank ; re-
quired the breadth of the brook, the heighth of the precipice
being 33 yards ? Ans. 56 feet.
1 6. A ladder of 50 feet long, being placed in a street, reached
a window 28 feet from the ground on one side ; and by turning
the ladder over, without removing the foot, it touched a mould-
ing 36 feet high on the other side ; required the breadth of the
street. Ans. 76. 1233335 feet.
17. Two ships sail from the same port; one, due east 84
leagues, and the other, directly south 135 leagues: how far are
they asunder I* Ans. 159 leagues.
PROBLEM V.
To find the circumference of a circle from its diameter.
RULE.
Multiply the diameter by 3 # 14159
OR
Multiply the diameter by 355, and divide the product by 113.
Ex. 1. If the diameter of the earth be 7930 miles, what is
the circumference ?• 3. 14159 X 7930 = 24913 miles.
2. How many miles does the earth move, in revolving round
the sun ; supposing the orbit to be a circle, whose diameter is
190 million miles? Ans. 596,902,100.
3. If the diameter of a wheel be Af feet ; what is the circum-
ference? Ans. 14 feet 1-J- inches.
* Note. The square toot may in the same manner be applied
to navigation ; and when deprived of other means of solving
problems of that nature, the following proportion will serve to
find the course.
As the sum of the hypothenuse (or distance) and half the
greater leg. (whether difference of latitude, or departure) is to
the less leg. so is 86, to the sine of the angle opposite the less
leg.
94 LOGARITHMLCK
4. What is the circumference of a circular island, whose
diameter is 45 rods ? Ans. 141 rods, 1^ yard.
5. What is the whole distance of space, through which the
planet Hershel moves, in revolving round the center of the sys-
tem, supposing its orbit to be a circle, whose diameter is 1,800
millions miles ? Ans. 11,309,724,000 miles*
PROBLEM VI.
To find the diameter of a circle from its circumference.
RULE.
Divide the circumference bv 3.14159
OR
Multiply the circumference by 113, and divide the product by
555 : Or multiply the circumference by .31831, and the product
will be the diameter.
Ex. 1. If the circumference of the earth be 24913 mile$,
what is the diameter ? Ans. 7930 miles.
2. If the periphery of a wheel be 6 feet 6 inches ; what is its
diameter ? Ans. 2 feet ■£? inches.
3. If the circumference of the Sun be 2,800,000 miles, what
is his diameter ? Ans. 891,267 miles.
4. If the circumference of the Moon be 6850 miles, what is
her diameter ? Ans. 2 180 miles.
t 5. If the whole extent of the earth's orbit be 596,902 % 10O
miles how far are we from the Sun ? Ans. 95,000,O00miles..
PROBLEM VII.
To find the Area of a circle.
RULE.
Multiply the square of the diameter by .7854
OR
Multiply half the diameter into half the circumference.
Ex. 1. What is the area of a circle whose diameter is 623 ?
Ans. 304836
2/ How many acres are there in a circular island, whose di-
ameter is 124 rods ? Ans. 75 acres, 76 rods.
- 3. What is the area of a circle, whose diameter is 7 feet.
Ans. 38.4846
,4. How many square ards yare in a circle, whose diameter is
Si feet?, Ans. 1. 069
5. What is the area of a circle, whose diameter is 1, and
whose circumference is 3. 14159 ? Ans. .7854
If the diameter of a circle is not given, the area may be found
by multiplying the square of the circumference by .07958. (Sup
Euc. 8. 1.)
. Ex. 1. What is the area of a circle, whose circumference is
1 36 feet ? A&5.- 1 472 feet;. .
ARITHMETIC*. *s
2. What is the surface of a circular fish pond, which is 10,
rods in circumference ? Abb. 7. 95800 rods.
PROBLEM VIII.
To find the diameter 9/ the earth, from the known height of a
distant mountain, whose summit is just visibUb in the horizon,
RULE.
•From the square of the height, subtract the height.
Ex. U The summit of Mount Chimhorazo in South America,
is about 4 miles above the level of the ocean. If a strait line
from this touch the surface of the water at the distance of 178£
miles ; what is the diameter of the earth ? Aas. 7940 miles.
2. The White Mountains in New-Hampshire are about 7100
feet high above the level of Connecticut River ; and a strait line
from the summit of the mountains will touch the surface of the
water at the distance of 103^ miles, what is the diameter of the
earth? Ans. 7940 miles,
PROBLEM IX.
To find the greatest distance at which a given object can bo
seen on the surface of the earth.
RULE.
To the product of the height of the object into the diameter
of the earth, add the square of the height, and extract the square
root of the sum.
Ex. 1. If the diameter of the earth be 7940 miles, and
Mount JEtna 2 miles high, how far can its summit be seen at
sea ? t Ans. 126 miles.
2. Suppose the diameter of the earth as in the first example*;
at what distance may a steeple be seen 9n level ground, allow-
ing it to be 1 65 feet in height ? Ans. 2 1-j. miles.
If a man standing on a level plain, has his eye elevated 5* feet
above the ground ; to what distance can he see the surface of
the plain ? Ans. 2-J miles.
4. The top of a ship's mast 132 feet high is just visible in the
'horizon, to an observer on the deck of another ship 33 feet from
the surface of the water ; how far are they asunder ?
Ans. 2 lj. miles.
•Note 1. See Euclid's Eliments, 36. 3*
Note. 2. The actual distance at which an object can be seen, is
increased by the refraction of the rays of light in the air. (See
End. Nat. Pkil.) But if no allowance be made for this refraction.,
the distance to which a person can see the plane surface of the
deean, is equal to a tangent to the earth drawn from the ob-
lervtr's eytf.
9t LOGABITHMICK
PROBLEM X.
Tojind the jfrea of a Triangle.
RULE.
Multiply the base of the given triangle into half its perpen-
dicular height ; or half the base into the whole perpendicular,
and the product will be the answer.
Ex. 1. Required the area of a triangle whose base, or longest
side is 36 inches, and the perpendicular height 16 inches.
Ans. 36 X 8 =? 289 inches.
. 2. Required the area of a triangular garden, whose base, or
longest side is 15.6 rods, and the perpendicular opposite the
base is 9 rods. .Ans. 70.2 rods.
PROBLEM XI.
Tojind the convex eur/ace of a Cylinder*
*Diffinition. A Cylinder is a round body whose bases are cir-
cles, like a round column or stick of timber of equal bigness
from end to end.
RULE.
Multiply the length into the circumference of the base.
Ex. 1 . How many square feet in the superficial contents of a
cylinder which is 42 feet long, and 15 inches in diameter.
Ans. 42 X 1.35 X 3.14159 = 164.933 square feet.
2. Required the convex surface of a cylindrical stick of tim-
ber, whose axis is 5 feet, and the diameter 7 inches.
Ans. 1520 inches.
PROBLEM XII.
Tojind the solidity of a Cylinder.
RULE.
Find the area of the base (by Prob. VII.) which multiply into
the length, and the product will be the solid contents.
1. What is the solid contents of a round stick of timber whose
diameter is 18 inches, and length 20 feet ?
18 in. = 1.5 ft.
Xl.5
2.25 X. 7854= 1.76715 area of base.
Or 18 inches 20 length.
18 inches. , Ans. 35*34300
324 X-7854 « 254*4696 inches, area ef the base.
20 length in feet.
• i J' « u ' ■■
144)508$-3920(35-343 solid feet, Ans.
2. What is the solidity of cylinder, whose length is 121, and
diameter 45.2 ? Ans. 45.3*X-7854X121 = 194156.6
ARITHMETIC*. 97
PROBLEM XIII.
tyfind the solidity of a CotfE.
Definition. A Con* is a solid yi hose ty*e is a circle, from
which it decreases gradually to a point in the top, called the
VERTEX.
A-line drawn from the vertex, perpendicular to the base, is call-
ed the height of the cone.
RULE.
Multiply jthe area of the base by the height, and £ of the
product will be the content.
Ex. i. What |3 the soliclity of a cone, whose height is. 1?
feet 6 inches, and the diameter of the base 2 feet 6 inches I
25* X-7854 X 12-5 ~ 3 = 20-453125 feet, Ans.
2. Required the solidity of a conical monument, that is 9 feet
high, and the diameter of its base 2£ feet.
Ans, 14-726250 feet.
PROBLEM XIV.
To find the solidity of a Frustrum of a eone.
Definition. A Frustrum of a cone is what remains after any
portion of the upper end is cut off, by a plane paralell to the
base.
RULE.
Add together the areas of the two ends, and the square root
of the product of these areas ; and multiply the sum by £of
the perpendicular height, and the result will be the solid con*'
tent.
OR
2. Divide the difference of the cubes of the diameters of the
two ends, by the difference of the dtameters,*and this quotient}
being multiplied by -7854 and again by \ of the height, will give
the solidity.
EXAMPLES.
1. Required the solidity of a frustrum of a cone, whose altitude,
or height is 18 feet, the greatest diameter 8 feet, and the least 4
{pet. By the 1st. Rule.
89 X -7854 = 50*2656, = area of base.
4* X -7854 =s 12*5664, = do. of the other erit.
yM 2*5664 X 50-2656 = 25*1328, = J the pro'd of the 2 areas..
Multiply 87*9648 the sum
by ^ of 18 ~ 6
Ans. 527*7888 solid inches..'
By the 2d. Rule.
.8?— ,4* s 448 -~ (8—4) = 112 X 'P854 % 6 =
52?-783$ ift. Ai>9.,
13
98 LOGARITHMICK
The latter method, i* many cases, will be found preferable tf
the former in point of expidition. •
2. .What is the content of the frustrum of a conical blocks
whose height is 20 inches, and the diameter of its two ends 28
and 20 inches ? Ans. 9131-5840^
The number of gallons or bushels which a vessel will contain
may be found, by calculating the capacity in inches y and then
dividing by the number of inches in 1 gallon or bushel ; as by
the following
TABLE OF SOLID MEASURE.
1728
cubic inches .
= 1 cubic foot^
27
cubic feet
= 1 cubic yard,
4492}
cubic feet
= 1 cubic rod,
32768000
cubic rods
= 1 cubic mile,
282
cubic inches
= 1 ale gallon,
231
cubic inches
= 1 wine gallon,
2150-42
cubic inches
= 1 bushel,
1 cubic foot of pure water weighs 1000 ounces, Avoir-
dupois, or 62| pounds.
EXAMPLES.
1. What is the capacity of a conical cistern, which is 9 feet
deep, 4 feet in diameter at the bottom, and 3 feet at the top I
Ans. 87*18 cubic feet X 7-4805* = 652*15 wine gallons.
2. How many gallons of ale can be put into a vat in the form of
a conic frustrum, if the larger diameter be 7 feet, the smalle
diameter 6 feet and the depth 8 feet ? Ans. 1886*5458 gallons.
3. There is a cistern in a distillery whose altitude is 10 feet
the greater diameter 14 feet, and the smaller diameter 12 feet,
required its capacity in hogsheads.
143— 12* — 14—12 X -7854 X *■*/ X 7-4805 -s- 63 =:
Ans. 157*918193 hhd.
PROBLEM XV.
To find the surface of a Sfihere.
JDeffimtion. A Sphere, or globe is a round solid- body, in
the center of which is a point, from which all lines dratorn to>
the surface are equal.
RULE.
Multiply the diameter by the circumference.
Note. In like manner, the convex surface of any zone or seg-
ment is found by multiplying its height by the whole circum-
ference of the sphere.
* Note. When the capacity is in feet, multiply by 7-4805, be-
cause 1 J|| = 7*4805 the number of wine gallons in 1 cubic food
When the ale gallon is required, multiply the feet by 6-1276,
because ■*} f f = 6-1276 ; but if the capacity be calculated in
inches divide by the number of cubic inches, in tfce gallon.
ARITHMETIC^ ♦ 4?
EXAMPLES.
1; What is the convex surface of a sphere, whose diameter
is i inches, and circumference 22 inches ?
: Ans. 7 X 22 =154 in.
2 Required the surface of a globe, whose diameter or axis,
is 24 inches, 24 X 3-14159 X 24 = 1809-5616 inches, Ans.
3. Considering the earth as a sphere, whose circumference is
2500$ miles ; how many square miles are there on its surface ?
An»i 198943750sq. miles.
4. The axis of a sphere being 42 inches, what is the convey
superficies of the segment, whose height is 9 inches ?
Ans. 42 X 3- 14156 X 9 = 1 187-5248 inches.
5. If the circumference ofthfe sun be 280000a miles, what i5
the surface?: Ans. 2495547600000 sq. miles.
PROBLEM XVI.
To find the solidity of a Sphere*
RULE.
1. Multiply the cube of the diameter by *5236.
OR
2. Multiply the square of. the diameter bf £ of the circum*
ference.
OR'
3. Multiply th« surface by \ of the the- diameter.
EXAMPLES.
1. What is the solidity of a sphere, whose diameter is 1 foot?
12? X -5236;= 904-7808. inches, Ans.
; Or 12» X 6-28318 = 904-7808 inches,
Or 455-38896 X 2 = 904-7808 inches^
2. What is the solid content of a sphere 4 feet 6 inches in
iameter ? Ans. 47.7 1305 00. feet.
3. Required the number of solid miles contained in the earth,
Supposing its circumference* to M& 2500D miles:""
Ans. 263858149120 miles.
4^ How many wine gallons will fill a hollow sphere 2 feet 8
inches in diameter ?
The capacity is 9.9288 feet X 7:4805,= 1 hhd.l 1.27 gallbns.
5. How many, gallons of water may be put into a hollow
sphere that is 4 feet in diameter, and what will be the weight of
the water ?
Note. The numbers 3*14159, *7*54, *5236, should be made
perfectly familliar. The first expresses the ratio of the cir-
cumference of a circle to the diameter ; the second^ the ratio of
the area of a circle to the square of the diameter ; and ^e.
third, the ratio of the solidity of a sphere to the cube of tlflih
ameter.
The second is £, and the third is £ of the first.
t6» LOtJAKITfiMICK
Amv 905.33832704 gallons, and the weight is 12833.6*544 lfei
6. If the diameter of the moon be 2180 miles, what is her
ablidity ? Ans. 5424600000 miles.
, When the solidity o{ a sphere is given, the diameter may be
found by dividing the solidity by .5236, and Extracting the cube
root of the quotient.
7. What is the diameter of a sphere, whose solidity is 6 $.45
cubic feet ? W'H'H ~* fi feet Ans.
8. What must be the diameter of a sphere, to contain 105f
gallons of wine ? Ans. 3 feet.
9. Required the diameter of a globe, to contain 16755 pounds
of water. Ans. 8 feet.
4 10. How imtny globes that are 3 inches each in diameter, are
equal to another globe whbse diameter is 12 inches ?
Ans. 64-
Note. The solid contents of similar figures are in proporti&n
to each other, as the cubes of their homologous* sides, or diam-
eters. Euc. El.
12. If a cannon ball 6 inches in diameter, weigh 33lb. what
Will another ball weigh, whose diameter is 3 inches ?
6 3 = 21^ and 3* = 27, then as 216 : 32 : : 27 : 4 lb. Ans.
13. If a metalic globe 8 inches in diameter, weigh 72 lb.
what will be the weight of a globe of the same foetal, whose di-
ameter shall be 4 inches ? ' Ans. 9 lb.
. 14. If a globe of silver 3 inches in diameter, be worth 8*50^
fibw nferoy such globes will be equal in value to £9600 JT
Ans. \4.
ANNUITIES, OR PENSIONS..
An Annuity, is a ium of money payable every year, for a
certain number of years, or forever.
When the debtor keeps the annuity in his own hands beyond
the time of payment, it is said to be in Arrears.
The sum of all the annuities for the time they have been for>
born together with the interest due upon each, is called the
Amount.
If an aftnuity be bought off? or paid all at once, at the begip.
tring of the first year, the price, which ought to be given for it,
is called the Present Worth.
To find the amount of an AnkVjty at Stm?le I^ter^St.
ARITHMETIC*. itfT
RULE.
1, Find the sum of the natural series of numbers, 1, 2', S, &ti
up to the giren number of years wanting tmc.
2. Multiply this sum by one year's interest of the annuity, and
the product will be the whole interest due upon the annuity.
3* To this product add the product of the annuity multiplied
Into the time, and the sum will be the amount sought.
EXAMPLES.
1. What is the amount of an annuity of £50 for 7 years, aT-
Ibwing simple interest at 5 pr. cent ?
1 +2 +34-4 4.54. 6 = 21=3X7
£2 10s. = 1 years interest of 50f.
3
7 lit
7
52 10
350 0=3 £"50X7
£402 10 = amount required.
2. If a pension of g600 pr. annum be forborn S years, what
% will it amount to, allowing 6 pr. cent simple interest ?
Ans. 83360
3* If a salary of 8750 annually, remain unpaid for 4 years,
how much mu£t be paid at the end of said term, allowing 4£ pr.
cent simple interest. Ans. $3202 50
Tojind the jiretent worth of an Annuity at Simfile Intercat.
RULE.
Find the present worth of each year by itself, discounting from
the time it becomes due, and the sum of all these will be the
present worth required.
EXAMPLES.
^ 1. What is the present worth of an annuity of glo6, to con-
tinue 5 years, at 6 pr. cent pr. annum simple interest ?
1 06"! T94.&39 6 = present worth for 1 st year.
112 j 89.2S57 =2 « 2d year.
128 U-: 100 : * 100 : ^ 84.7457 «s " 3d year.
124 180.6451= ** 4th year.
J30j \J6.92S0 a: « *th year.
425.9391 a= 35425 93 CtS. 9m. T ^
present worth of the annuity required.
2. What is the present wtfrth of $400 pr. annum, to continue
4 years at 6 pr. cent ? Ans. 81396.065^3
j
tar LOGARITHMICK
3. What is the present worth of an annuity, or pension of
jf 500, to continue 4 years, at 5 pr. cent pr. annum, simple inter-
est? Ans. f 1782 3s. 8|d.
To find thh Amount of an Annuity at Compound interest.
RULE.
1. Make 1 the first term of a geometrical progression, and
the amount of £1. or SI for one year, at the given rate pr. cent,
the ratio.
2. Carry the series to as many terms as the number of years*
and find its sum. v
3. Multiply the sum thus found by the given annuity, the
product will be the amount sought.
EXAMPLES.
i. If a salary of g600 be forborn (or remain unpaid) 7 years ;
what will it amount to at 6 pr. cent pr. annum, compound inter-
est ? 1 + 1 .060000 J- 1.123600 +1.19101 16+1.262476 + 1.338225
►1-1.418519+1.503630= 8.897466 = sum of the series.*
Multiplied by 600
gives 5338.4796 = g5338 47 cts. 9/ ff m* the amount
sought.
Or, By Table III.
Multiply the Tabular number under the rate, anjj opposite to
the time, by the annuity, and the product will be the amount
sought.
2. If a pension of g 175 pr. annum, be forborn 20 years, at 6
pr. cent compound interest ; what is the amount ?
Tabular number = 36.785590
175 = Annuity.
An> 6437.478250 = 86437.47 C. 8 J m.
3. Suppose g 50 pr. annum, with compound interest at 5£ pr.
cent.be 10 years in arrears ; required the amount.
Tabular number = 12-875354 X 50 = g 643-76c.7m. Ans.
4. What will a pension of £ 120 pr. annum, amount to in 3
years, at 5 pr. cent, compound interest ? Ans. £378 6s.
5. The salary of the President of the United States, is g 25000,
supposing the whole be in arrears during the period of his elec-
*The sum of the series thus found, is the amount of £l 9 or
gl annuity, for the given time, which may be found in Table
I Hi ready calculated. (The method of constructing these To*
dies vnll be shown hereafter.) y
Hence, either the amount, or present w^orth of annuities;
continuing for a term not exceeding 40 years, may readily be*
found by the Tables for that purpose. '
ARITHMETIC**. 1Q3
tSJm, or 4 years ; what would then be the amount of his salary,
allowing 6 pr. cent, compound interest ? Ans. g 109365,40
To find the present worth of Annuities at Compound Inter est '•
RULE.
1. Divide the annuity by the ratio, or the amount of g l,er
£\ for one year, and the quotient will be the present worth of
the first year's annuity.
2. Divide the annuity by the square of the ratio, and the quo-
tient will be the present worth of the annuity for the second
year.
3. Find, in like manner, the present worth of each year by
itself, and the sum of all these will be the present worth of the
annuity sought.
EXAMPLES.
1. What is the present worth of an annuity of g 40 to con-
tinue 5 years, discounting at 5 pr. cent. pr. annum, compound
interest ?
Ratio) 1 :-: 1-05)40* 00000(38095= present worth 1st. year.
Ratio)*— 1- 1025)40- 00000(36-281 = « 2d. year.
Ratio)*-- 1-57525)40' 00000(34-556 = « 3d. year.
Ratio)* — 1-2 15506)40- 00000(32 899= « 4th. year.
Ratio)* = 1-2762 18)40. 00000(31-342 = " 5th.year.
173173= g 173 17c. 3m. whole
present worth of the annuity required.
Or, By Table IV.
Multiply the tabular number under the rate and opposite to)
the time, by the annuity, and the product will be the present
worth required.
2. What is the present worth of an annuity of g 50 to con-
tinue 5 years, at 6 pr. cent, pr. annum, compound interest I
Tabular number = 4-21236 X 50 = Ans. g 210 61 8
3. If the pension of an officer, serving in the Revolutionary
War, be 20 dollars a month, or 240 dollars annually ; what is
-its present worth, allowing a discount of 6 pr. cent, pr. annum,
compound interest, supposing it to continue 10 years ?
Ans. g 1766 41c. 9^m.
To find thefire8ent worth of a Freehold Estate, or an
Annuity to continue forever, at Compound Interest.
RULE:
As the ratepr. cent, is to g 100, so is the yearly income u
the value required.
I. What is the worth of a freehold estate of g 40 pr. annui*
allowing 5 pr. cent, to the purchaser ?
As 5 > 100 : j 40 : g 800 Ans.
1*M LOGARITHM1CK
• *
a. What >• a freehold estate of £7§ a year worth, allowing
the buyer 6 pr. sent, compound interest for bis money ?
Ans. £ 1250
3. An estate brings in yearly 8 79*2a what would it sell for, al-
lowing the purchaser 4| pr. cent, compound interest for his
money ? Ans. 81760.
To find the fir ceent worth of an Annuity, or Freehold E*ta&
in Reversion at compound interest.
RULE.
Find the present value of the estate (by the foregoing rule)
as though it were to be entered on immediately, and divide said
value by that power of the ratio denoted by the time of rever-
sion, and the quotient will be the present werth of the Estate
in Reversion.
EXAMPLES.
1 . The reversion of a freehold estate of £ 79.4s. pr. annum, to
commence 7 years hence, is to be sold ; what is it worth in
ready money, allowing the purchaser 4} pr. cent, for his money?
As 4-5 : 100 * .: 79-2 : 1760 » present worth, if entered on
immediately.
And 1045> 7 = 1- 360862) 1760-0d0( 1203*297 = £ 1293. 5s.
1 1 Jd. = present worth of 17601. for 7 years, or the whole pres*
ent worth.
Or, By Table IV.
Find the present worth af the annuity, or rent, for the time of
reversion, which subtract from the value of the immediate pos-
session, and the remainder will be the value of the estate in
reversion.
2. What if the present worth of a freehold estate of g 40 pr.
annum, to commenpe 7 years hence allowing the purchaser 5
pr. cent ?
Tabular number = 5-78-637
40 at annnky, or rant. .
231*45480 =c present worth of rent.
& : ,100 : : 40 ; 8000000 = value of immediate possession.
5685452 as g 568- 54c. 5|m. An*.
3. Whkh is the most valuable, an income of £200 pr. amium
Br 15 years, or the reversion of an equal income forever after*
ward, computing at the rate of 5 pr. cent, pr. annum, compound
ihterest ?
Ans. The first term of 15 years is better than the reversion
fflreyer afterward, by 17/. 18s.7*d»
ARITHMETICS 105
VULGAR FRACTIONS.
The learner will find it convenient, and indeed necessary, as
lie progresses in the more intricate parts of Arithmetick, to be
thoroughly'acquainted with Vulgar fractions. They were brief-
ly introduced immediately after Simple Proportion (page 56) as
a preliminary to the subject of Decimal Arithmitick ; but we
shall here consider, more extensively, the intimate relation be-
tween Vulgar, and Decimal Fractions, together with their rela-
tive and important use in Arithmetick in general.
Vulgar Fractions are either proper, improper, simple,
compound, or mixed.
1. A Simple , or proper fraction is one, whose numerator is
less than the denominator ; as f ,£,£, Sec.
fc. An Improper fraction is one, whose numerator exceeds the
denominator ; as f ,}|, Sec.
4. A Mixed JVumder is composed of a whole number and
a fraction ; as 81,25^, &c,
Note. Any whole number may be expressed like a fraction,
by writing. \ under the givem number for a denominator j as
8= f and 12 = y, &c.
3. A Compound fraction is the fraction of a fraction, coupletf
by the word of as ; | of J of i of y, Sec.
5. The Common Measure of two or more numbers is that
, number, which will divide each of them without a remainder.
Thus, 4 is the common measure. of 12, 16, and 20 ; and the
greatest number^ that will do this, is called the Greatest Ctrnmon
Measure.
6. A number, which can be measured by two or more num-
bers, is called the Common Multiple ; and if it be the least
number, which can be so measured, is called. their Least Com-
mon Multiple : thus, 24, 36, 48 and 60 are each a common muly
tiple of 3, 4, and 6 ; but their least common multiple is 12.
To find the least Common Multiple of two or more numbers.
RULE.
1. Divide by any number that will divide two or more of the
given numbers without a remainder, and set the quotient| to-
gether with the undivided numbers, in a line beneath.
2. Divide the second lines as before, and so on till there are
no two numbers that can be divided ; then, the continued pro-
duct of the divisors and quotients, will give the multiple re-
quired.
14
106 LOGARITHMICK
EXAMPLES.
l. What is the least common multiple of 6, 8, 10, and 12.
2)6 8 10 12
Thus.
3)3 4 5 6
2)1 4 5 2
12 5 1
The product of the divisors = 12 and 12X2 X 5 120 Ans.
2. What is the least common multiple of 4 and 63 ?
Ans. 12
3. What is the least common multiple of 3,4,8, and 12 ?
Ans. 24
4. What is the least common multiple of 4,5,6, and 10 ?
Ans. 60
5. What is the least number that can be divided by the 9
digits, separately, without a remainder ? Ans. 2520
REDUCTION OF VULGAR FRACTIONS:
Is the bringing them out of one form into another, in order to
prepare them for the operations of Addition, Subtraction, fee.
CASE I.
To abbreviate y or reduce fractions to their lowest terms*
RULE.
1. Divide the terms of the given fraction by any number, that
will divide them without a remainder, and these quotients again
in the same manner ; and so on till it appears, that there is no
number greater than I, which will divide them again, and the
fraction will be in its lowest terms.
OR
2. Divide both the terms of the fraction by their greatest
common measure, and the quotients will be the lowest terras of
the fraction required.
EXAMPLES.
1. Reduce ift to its lowest terms.
(2) (2) (3) (2) (2)
*H = t¥* ^U = H = A «■? the answer.— (see page 56)
ARITHMETIC*: JOT
Or thus; 144)240(1
144
96)144(1
96
48)96(2
48
Therefore 48 is the greatest common measure, and 48)^|f = J
the same as before.
2. Reduce ffff to its lowest terms. Ans. f
3. Reduce Jf $$ to its lowest terms. Ans. £
4. Reduce §£f to its least terms. Ans. -^
5. Reduce £$f to its lowest terms. Ans. ££
CASE II.
To reduce a Mixed Number to its equivalent improper fraction!.
RULE.
Multiply the whole number by the denominator of the frac-
tion, and add the numerator to the product, then that sum writ-
ten above the denominator will form the fraction required.
EXAMPLES.
1. Reduce 25f to its equivalent improper fraction.
25X8 + 3 = 193, then the fraction will become « | c
2. Reduce 27f to its equivalent improper fraction.
Ans. *$*
3. Reduce 45 J to its equivalent improper fraction.
Ans. 3|7
4. Reduce 100$f to its equivalent improper fraction.
Ans. *f|9
5. Reduce 15|$ to its equivalent improper fraction.
Ans. « T »
CASE- III. •
To find the value of an improper fraction.
RULE.
Divide the numerator by the denominator, and the quotient
will be the whole or mixed number sought.
EXAMPLES.
1. Find the value of 3 |f 8 21)38.48(183^ Ans.
2. Find the value of 'A 8 " Ans. 9.
3. Find the value of x i$ s Ans. 56 U
4. Reduce * r y to its equivalent whoje, or mixed number.
Ans. 84 &
5. Reduce 4 |^ f to its equivalent whole, or mixed number.
Ans. 173 A
CASE IV.
To reduce a whole number to an equivalent /ractuta* hvovnfc a
given denominator.
10S LOGARITHMICK
RULE.
Multiply the whole number by the given denominator, and
place the product over the said denominator, and it will form
the fraction required.
EXAMPLES.
1. Reduce 8 to a fraction, whose denominator shall be 9.
8X9 = 72 ; and the fraction will become V Ans «
2. Reduce 13 to a fraction, whose denominator shall be 12.
Ans. Vt
3. Reduce 100 to a fraction, whose denominator shall be 79.
Ans. 7 ^°
CASE V.
To reduce a compound fraction to an equivalent simple one.
RULE.
1 . Reduce all whole and mixed numbers to their equivalent
fractions.
2. Multiply all the numerators together for a new numerator,
and all the denominators together for the denominator, and they
will form the fraction required.
EXAMPLES.
1. Reduce 4 of £ of £ of f- M to a simple fraction.
2X3X4X8= 192
■ ■ ■■ m — = j-f the answer.
3 X 4X5 X 11 =660
2 . Reduce £ of 4 of £ of ^ to a simple fraction. Ans. ^
3. Reduce i of £ of 10 to a simple fraction. Ans. 4 J
4. Reduce 4 .oi f of & of 2 If to a simple fraction.
Ans.2ifi*
CASE VI.
To reduce fractions of different denominations to equivalent
fractions^ having a common denominator.
RULE.
1 Reduce all fractions to simple terms.
2. Multiply each numerator into all the denominators, except
its own, for a new numerator ; and all the denominators togeth-
er, for a common denominator, which written under the several
numerators, will give the fractions required.
EXAMPLES.
,1. Reduce \% s and $ to equivalent fractions, having a com-
mon denominator.
1 X 5 X 7 = 35 the new numerator for i
3X2X7 = 42 ,: « for f
4X2 X 5 = 40 « for 4
2X 5 X 7 = 70 the common denominator.
Therefore the new equivalent fractions are ^f, 4o» an( ^ H
2. Reduce J, T V and }£ t0 a common denominator.
ARITHMETICS tO»
3. Reduce |, J, and 1 2£ to a common denomlnatoK
4. Reduce |, f of J, 5t> and^ to a common denominator.
5. Reduce |,| f £, 7£* and ^ to a Common denominator.
Ans. tfrVtHt'Wtf'tfk
CASE VII.
To reduce the fraction of one denomination to tjie fraction of
another, retaining the same value-.
RULE.
1 . Reduce the given fraction to such a compound one* as will
express the value of the given fraction, by comparing it with ail
the denominations between it and that denomination to which it
is to be reduced.
2. Reduce the compound fraction) thus made, tp a simple
one. (See Case V.)
examples. ,
1. Reduce $ of a penny to the fraction of a pound.
By comparing it, it becomes £ of T V of T J of a pound.
3X1X1
- i v»- = ytfVy the answer.
8 X 12X20
2. Reduce T ^ of a pound to the fraction of a penny*
Make a compound fraction of it thus ;
TtfW of V of Y = tW* = * the Ans.
3. Reduce f of a shilling to the fraction of a pound*
Ans* J*
4. Reduce f of a farthing to the fraction of a pound*
Ans. r J in
5. Reduce f of a pound avoirdupois to the fraction of a cwt.
* ■ \ Ans* - ¥ f T
6. Reduce T7 \ T of a hhd. of wine to the fraction of a pint.
Ans. ^
7. Reduce S\ furlongs to the fraction of a mile* Ans. |j-
8. Reduce T ^ 7 of a week to the fraction of an hour* Ans. $
9. Reduce 7s. 6d. to the fraction of a pound. Ans. $
10. Reduce 5}d. to the fraction of a shilling. Ans. }|
ADDITION OF VULGAR FRACTIONS.
RULE*
Reduce compound fractions to single dnes ; mitced numb*
to improper fractions ; fractions of different integers to thot
of the same ; and all of them to a common denominator : tht
15
J
110 LOGARITHMS*.
the sum of the numerators written over the common ^ieflominar
tor, will be the sum of the fractions required.
BKAMPI&&.
Add 3fr, $, | of }, and 7 together.
First 3f « y,4of£ = fti-r ** J
Then the factions are y , J , T ^, and *. Xherefore,
29X8XlOXl=a 93*20
7 X8 X 10 X 1 «= 560
7 X 8 X 8 X I === 448
7 X8 X 8X10 =3=4480
ir
7808
— «= 12} the answer
8X8X10X1= 640
2. Add £, ^, and £ of 5* together. Atis. ^> n
3. Add 12 J, 3§, and 4 J together. Ans. 20 ji
4. Add $1. Js. and yj of a penny together.
Ans. 3s* id. i|fqrs.
5. What is. the sum of $ of 15/. 3f/. £of f of f«f a pound,
and | of ^ of a shilling ? Ans. £i 17s. 5|d.
6. Add | of a mile, § of a yard, and J of a foot together.
Ans. 120 rods, 2 ft 9 in.
7. Add 4 of a ton, and ^ °f cwt « together.
Ans. 12«wt. 1 qr.8lb. 12^.«b.
8. Add a of a week, { of a day) and $ of atn hour together.
Ans* 2d* 14h. #0m.
SUBTRACTION OF VULGAR FRACTIONS.
RULE.
Prepare the fractions as in addition, and the difference of the
numerators, written above the common denominator, will gige
the difference of the fractions required.
EXAMPLES.
1. Fromf take f of -f.
i of ^ = ^r ; then the fractions are ^ r and f, therefore
2 X 3 a 6 ("and 42 — 6= 36
21 X 2 = 42 J Ans. f f = $
and 21 X 3 = 63 the common jdenomihator. £
2. From f& take ^. , Ans. U&
3. From 14^ take ^ of 19. Ans. 1 T 7 T
4. From \ L take £ *. Ans. 9s. 3d.
5. From -f oz. take -J of a'pwt Ans. 1 1 pwt. 3gr.
ARITHMETICS in
*. From 3$ weeks, take \ of a day, «nd i of * of $ of an hour.
Ans. 3vr.4d. I2h. 19m. 17^ sec.
7. The sum of three number* is 56} ; thejirst number is 12-J
atid the second 2l T ^ ; required the Mzrof. Ans. 22 T V T
8. What number added to 1 1 4 will make 36$f % ?
Ans. 24ft.
MULTIPLICATION OF VULGAR FRACTIONS,
Reduce compound fractions to single ones, and mixed num-
bers to improper fractions ; then multiply all the numerators to-
gether for a new numerator, and all the denominators together
for the denominator of the product required.
EXAMPLES.
1. Multiply i of 7 by I - Ans> 11
2. Multiply \ by \ Ans. «J.;
3. Multiply 7i by 9J Ans. 69f
4. Multiply I of i by £ of 3£ Ans,-^
5. Multiply 4i \ of i and 18 f continually together.
Ans.ft^ ?
DiyfSION OF VULGAR FRACTIONS.
RULE.
Prepare the fractions as in .multiplication ; then invert the
djyisor, and proceed exactly as in multiplication 1 the product
Twill be the quotient required.
examples.
1 . Divide f of 9 by f of 7\
2 X 9
j of f = — .. ^ V **& $ of H—Hl therefore,
5X1
18X16
V -5-if = " *= 4*1 Wy the quotient required.,
5X45
2. Divide f by §. Ans. I/ r
£. Divide \ by 4. Ans. ^V
- - - - Ans. 2 ? V
Airs. )8*£
4. Divide 4$ by f of 4,
5. Divide 7 by >.
112 LOGARITHMICK
6. Divide | of 1 9 by | of |. Ans. 7f
7. Divide i of f by | of }. Ans. f
8. What number multiplied by f , will make 1 1^ ?
Ans. 26$f-
SIMPLE PROPORTION IN VULGAR FRAC-
TJONS.
RULE.
1. Prepare the fraction as. before ; then state the question,
agreeably to the rules in Simple Proportion of whole numbers.
2. Consider whether the proportion be Direct or Inverse ; if
direct, then invert the Jirst term of the proportion ; but if. the
proportion be inverse, invert the third term.
3. Then multiply all the three terms continually together,
and the product will be the answer.
examples.
1 . If | of a yard cost f of a pound, what will j| of a yard cost ?
Thus, | ; £ : : || : jfj = 12s. Id. 2^ qrs. Ans.
2; If | of a yard cost ffof a pound, what will 91 yards cost ?
Ans. £4 10s. 2d. 2|qrs.
3. If Sd.buy j of a pound of sugar, how much will 10£d. buy T
Ans. lib.
4. At 7s. Od. If | qrs. pr. bushel, what will be the value of 15
bushels ? £5 5s. ?d. 0£ qr.
5. If | of a ship be worth $ of her cargo valued at 8000/. what,
is the whole ship and cargo worth ? Ans. £ 10031 Hs. ll T \-d.
6. A- and B. own a ship and cargo worth 16000/. A owns
J of the cargo and ^ of the ship ; but by accident at sea, they
lose \ of the cargo, which is 214f/ t less than B's. share in the
ship, required the values of the ship and cargo, and each one's
respective share in the same.
Ans. j£*4000 value of ship, and 12000/. do. of cargo.
A's share of cargo = 3937-^. C B's share of cargo = 6562*/.
A's do. of ship = 22854/. C B 's do. of ship = 17l4f/;
A*s do. of the whole £622&ft B's do. of the whole £8276}J
ARITHMETICS tis
REDUCTION OF DECIMAL FRACTIONS.
CASE I.
To reduce numbers of different denominations to their equiva-
lent value.
RULE.
Bring the givtn denominations 10 a vulgar fraction, and reduce
said fraction to its equivalent decimal value. (See Case lily
page 57, also Rule 2, page 64.)
EXAMPLES.
1. Reduce 10s. 6d. 2 qrs. to the decimal of a pound.
1/. X 20 X 12 X 4 « 960 and 10s X 12 X 6 X 4 -h 2 = 506,
therefore, 506 — 960 = .527085 Ans.
2. Reduce 13s. 5£d. to the decimal of a pound. Ans. .6729
t 3. Reduce 3 qrs. 2 na. to the decimal of a yard. Ans. t 875
4. Reduce 17 yds. 1 ft. 6 in. to the decimal of a mile.
Ans. .00994318
5. Reduce 10 weeks, 2 days, to the decimal of a year.
Ans. .1972J602, &c.
CASE IL
To .find the value of a decimal in the known parts of the integer.
RULE,
1. Multiply the decimal byj.he number of parts in the next
less denomination, and cut off so many places for a remainder,
on the right, as there are places in the given decimal.
% Multiply the remainder by the next inferiour denomina-
tion, and cut off a remainder as before ; and so on through all
the parts of the integer, and the several denominations stand*
ing on the left, make the answer.
EXAMPLES.
1. What is the value of .37623 of a pound I
20
7.42460
12
1.18080 Ans. 7s. 6d. 1 qr.
2. Wl,at is the value of .83229 1 6 of a pound ? Ans. 1 6a. 7-J6\
3. What is the value of .625 of a shilling ? Ans. 7£ck
4. What is the value of .76442 of a pound Troy?
Ans. 9 oz. 3 pwt, 11 gr.
£ Find the value of .875 of a yard. Ans. 3 qr. 2 na.
ll> LOGARITHMIC*
6.J Find the value of .61 of a ton of wine*
Ans. 2 hhds. 27 gals. 2 qts. 1 pt.
7. What is the value of .8469 of a degree I
Ans. 58 m. 6 fur. 35 po. ft. II in.
i* JVhat is the value of .569 of a year ?
Ans. 207 d. 16 h. 26 m. 24 sec.
CASE IV,
To find the decimal of any number of 'shillings^ pence , and far-
things , by jns/iection.
RULE.
1. Wrjte half the greatest even number of shillings for the-
first decimal figure.
2. Let the farthings in the given penee and farthings, possess,
the second and third places ; observing to increase the second
place by 5, if the shilings be odd; and the third place by 1, whea
the farthings exceed 12, and by 2, when they exceed 36.
EXAMPLES.
X. find the decimal expression of 9s. 7±d. by inspection-
.4 = f 8s.
.05 = for the odd shilling
SO = the farthings in 7\jL
1 for excess of 12:
£\4Sl ^ decimal required.
2. What is the decimal value of 17s. 8|d.? Am. •/-.8B$
3. What is the decimal expression of 7s. 9Jd. ? Ans. J.S91
4. Find the decimal value of 151. 3s. 9*d. Ans. £15*19
CASE V.
To find the value of any decimal of a pound by inspection*
RULE.
1. Double the first figure, or place of tenths in the decimal,
for so much of the answer in shillings, increasing the sum by 1,
if the second figure be 5, or more than 5*
2. After the 5 is deducted, call ,the remaining figures in the
second and third places, so many farthings, for the remainder of
the answer, abating I , if they exceed 12, and 2, if they exceed 36.-
Note. When the decimal has but 2 figures* if any thing re*
main after the shillings are deducted, annex cyphers on th*
right.
EXAMPLES.
i. Find the value of £.876, by inspection.
.876
16s. = double of 8.
1 5 for the 5 in the second place which*
£is to be deducted out of 7.
And 6 f= 26 farthings remain to be added~
Deduct i for the excess of 12:
Ans. 17s. 6 £d.
ARITHMETICS. 115
2.. Find the value of/\679, by inspection. Ans. 1 3s. 7d.
S. Find the value of/\842i by inspection. Ans. 16s. lOd,
4. Find the value ofjf.790 by inspection. Ans. 15s. 9£d.
5. Find the value of £.097 by inspection. Ans. Is. Hid.
SIMPLE PROPORTION BY DECIMALS,
RULE.
Seduce fractions to decimals, and state the question as in
whole numbers ; multiply the second and third terms together,
*nd divide by the first, and the quotient will be the answer.
EXAMPLES.
1. If | of a yard tost -f of a pound, what will 9-1 yard cost ?
f = .375 yds.
£ = .41.
9| = 9.625 yds.
"Therefore .375 yds. : 41. : : 9.625 yds. : £10.2666 ; or 10L 5s. Sd.
3 qrs.
2. If j. of a yard cost T 7 ¥ of a pound, what will T ^- of an English
*ll cost I £ of a yard = £ of f of f =^| of an ell.
Then f-f ell : T y. : ; T " r ell. : 44L == 9s. 8d. 2 qrs. Ans. or, '
« = 48 f
7 7 j = .5833 -J Then .48 : .5833 : : .4 ; .486 = 9s. 8d. £rs.
3. At 7£d. pr. lb. what will be the price of an cwt. of sugar ?'
7-5 X 1 12 = 840 = 3/. 10s. Ans.
4. What is the value of 3% cwt. of coffee at 23j cts. pr. lb. \
Ans. g98 70c.
5. What is the value of 2 qrs. 1 na. of velvet at 19s. 8f d. pr.
English ell ? Ans. 8s. lOd. Iqr. T \
6. If \ of a yard of satin cost 7s. 3d. how many yards can 1 buy
Jor 13/. 15s. 6d. ? Ans. 28f yds.
7. What is the value of £ of a tun of wine, when £ of a gallon
x;osts I of a pound ? » Ans. £\40
8. At li/« P r « c ^t. what does o-Jlb- come to? Ans. iO$d.
S. What is the tax upon 745/. 14s. 8d. at 3s. 6d. on the pound ?
Ans. £130 10s. 3}qrs.
10. A person shares f in a certain prize, and sells f of it for
171/. what was the whole amount of the prize ? Ans. £3$Q
1 1. If, when the days are 13f hours long, a traveller perform
his journey in 35£days ; in how many dayswiH he perform the
same journey, when the days are 1 l-j\ hours long ?
Anfc 40A.?4 <h*vs.
116 LOGARITHMICK
12. A regiment of soldiers, consisting ©f 976 men, are tojbe
new clothed, each coat to contain 2J yds. pf cloth, that is If yd,
wide, and to be lined with shalloon, } yd. wide ; how many yards
of shalloon will line them ? Ans. 4531 yds- 1 qr. 2^ na*
SIMPLE INTEREST BY DECIMALS:
A TABLE OP RATIOS.
RATE PER CENT. | RATIO, | RATE PER CENT. f RATI<£ |
4 I
3 t -03 \ Si
.04 | 6
4± t -045 I 6i
5 I -05 I 7
06 j
065 I
or I
Ratio is the simple interest of £ J, or g I for one year at the .
rate per cent, agreed on.
RULE
Multiply the principal, Ratio and Time continually together,
and the last product will be the interest required.
EXAMPLES.
1 . Required the interest of 537 dolls, 58 cts. for 4 years artd 6
months, at 5 per cent. pr. annum, simple interest.
55537.58 principal
.05 ratio.
26.8790 interest for 1 year.
4-5 multiply by the time*
1543950
1075160
8 1*0.95550 Anjs. g 120 95 cts. 5 m. &c.
2. WJiat is the interest on 8268 17 cts. for 3 years and 9«,
months, at 44 pr. cent, simple interest ?
Ans. 268.17 X .045 X 3.75 = 45.2536875=845 25 cts. 3 m. -
3. What is the interest of 1181. 9s. for 1 year and 6 month
at 6 pr. cent pr. annum ? Ans. 101. 13s. 2d. 2 qrs.
4 Required the amount of 648 dolls. 50 cts, lor 12.75 years
at 5i pr. cent, simple interest. Ans. &l 103 26 cts. -f-
5. What is the amount of 691. 8s. for 3 years, 1-J months, at
£ pr. cent, simple interest ? Ans. £h 1 6s. 6cl. 3 qrs
ARITHMETICS* ,%V
CASE II.
Tkc amount 9 time, and ratio giv<?n, to find the principal.
ROLE.
Multiply the ratio by the time, add unity to the product for a
divisor, by which divide the amount, and the quotient will be
the principal.
EXAMPLES.
1. What principal will amount to 264.3125, in 5 years, at 5
pr. cent pr. annum ?
.05 X 5 ± 1 = 1.25) 264.3125 (211.45 Ans.
2. What principal will amount to 2658.53550, in 44 years, at
S pr. cent. I . Ans. £537.58 cts.
3i What principal will amount to 8313.423687$, in 3 years
and 9 months, at 4$ pr. cent? Ans. 8268 17 cts.
4. What principal will amount to £956 10s. 4.125d.in 8 years
and 9 months, at 5§ pr. cent ? Ans. £645 15s.
CASE III.
The amount, principal, and time given, to find the ratio.
RULE.
Subtract the principal from the amount, divide the remainder
by the product of the time and principal, and the quotient will
be the ratio.
EXAMPLES.
i. At what rate pr. cent, will 8950 75 cts. amount to 31235.
9750, in 5 years ?
From the amount = 1235.9750
Take the principal = 950.75
950.75 X 5 = 4753.75 )285.2250(.06 =* 6 pr.cent. Ans.
285.2250
2. At wLat rate pr. cent will 87 1 5.45 cts. amount to 894.54940,
in 2 years and 5 months ? Ans. 6 pr cent.
3. What rate pr.ct. will 8268. 17 cts. amount to 83 1 3.4236875,
.in 3£ years ? Ans. 4-J pr. cent.
4. At what rate per cent will £319 5s. amount to £ 62.253750
in 3 years and 3 months ? Ans. 6 per cent.
CASE IV.
The amount, principal, and rate per cent given, to fipd
the time.
RULE
Subtract the principal from the amount, divide the remainder
by the product of the ratio and principal 5 and the quotient will
be the time.
EXAMPLES.
1. In what time will 8537 58 cts. amount to 8658.53550, at 5
pr. cent pr. annum ?
16
US JUOGARITHMICK r ^
From the amount 2653.55550
Subtract the principal 537.58
537.58 X ,05 26.8790) 1 20.95550(4^ year* answer,
120.95550
2. In what time will 8268 .17 cents amount to 8313.4236875,
*t 44 per cent per annum ? Ans. 3 years 9 months.
3 In what time will 8950 75 cents amount to 81235.9750, at
6 percent per annum? Ans. 5 years.
4. In what time will £ 319 5s. amount to £381.503750, at 6 per
cent ? Ans. 3£ years.
TO CALCULATE INTEREST FOR DAYS.
, RULE.
Multiply the principal by the given number of days, and thai
product by the ratio ; divide the last producjt by 365, and the
quotient will be the interest required.
EXAMPLES.
1. What is the interest of 1781. 15s. for 87 days, at 6 per cent
per annum ? 178.75 X .87 X .06 = 836*5500 -~- 365. = 2.2891
= 21. 5s. 9d. 1 qr. Ans.
As the process of division is generally more tedious than that
Of multiplication, it is often convenient to substitute one for the
other.
Now if we substitute, in the place of 365, the reciprocal of
that quality ; that is, the quotient arising from dividing 1 by
365, we shall have a number which multiplied mto the principal,
will give the same result, as dividing by its corellative quanti-
ty ; Thus, 1-s- 365 = '00274, therefore to multiply a given,
number by '00274,* is equivalent to dividing the same number
by 365.
2. What is the interest of 8 100, for 75 days, at 6pr. cent pr.
annum ? 100 X 75 X -06 X -00274 =81 23c. 3m. Ans.
3; What is the interest of 8 148 50 cts. for 96 days, at 5 pr.
cent pr. annum \ Ans. 8 1 95c. 3m.
4. What is the interest of 8 3 1 2 for 25 days, at 7\ pr. cent ?
Ans. 8 1 60c. 3m.
5. What will £V5 amount to in 256 days, at 4^ pr. cent pr.
annum? Ans. £5 12s li<L +
%!..* Note. The cyphers on the left may be always neglected, by
observing to point off 5 figures of the product, on the right for
decimal parts.
ARITHMETICS
U'9
i
<p-6~e^ ~ e^^^~^*-&-*-*>- * fy>--** »■ » ' e "" b' " *' w»« " »
TABLE I.
Shewing {he A-
5 mount of 1 Dol :
J lap, op 1 Pound for
i 3 1 Year*, at 5, and
16 pp. Cent. Simple
\ Interest.
) Years. 5
TAB LE II.
Shewing the
Rebate of 1 Dol-
lar, op 1 Pound for
31 Year 8, at 5 an<<
6 pr. Cent. Sim-
ple Inter eat.
%
1
105
2
110
3
MS
4
1-20
5
1-25
_.
6
1-30
7
I'35
8
1-40
9
1-45
10
1-50
,
— — .
11
1-55
12
1-60
IS
r-65
14
1-70
15
1-75
» ■
■■"■■—
16
1-80
17
1-85
18
1-90
19
p95
20
fc-00
■ '-
_
21
2-05
22
2*10
23
2-15
24
5-20 •
25
2.25
26
2-30-
27
2-35
28
2-40
29
' 2-45 |
30
2-50
31
2.55 1
2-26
2*321
9523SO
90909 1
•869565
•833333
'800000
•769230
•740740
•714286
689655
•666666
•645161
•625000
•606060
•588235
•571448
•555555
.540540
•526315
•512820
•500000
487804
■476190
'465116
454545
444444
434781
'425532
•416666
■408163
400000
•393157
892857
847457
806451
769250
735294
704225
675675'
649350
625000
602409
581395
561797
543478
526315
510204
495049
•480769
467289
454545
442477
431034
•420168
409836
400000
390625
381679
73134
364963
357143
34965C|
TABLl, i ll.
Shewing the Amo.i
of 1 Dollar, pr 1 Pour
for Months at 5, and t-l
pr. Cent. Simple Inte\ •
cat.
Mont
1-00416
100833
I 01249
1-01666
1*02083
1 02499
1 02916
103S33
1-03749
1-04166
1-04583
1 05000
1-005
1-0 l€i
1.015
1-020
1-025
1-030
1 055
1-040
1045
1-050
1*055
1-060
1
2
3
4
5
6
7
8
9
10
11
12
Shewing the Rebau '
or Present worth of l [
Dollar, or 1 Pound, for 1
Months dicouniing at?
5, and 6 per Cent. SiVw-T
file Inter eat. Y
Monti.*
99585
-99173
■98766
98361
97959
•97561
•97263
■96772
•96387
•96006
95617
95238
•99502
.99009
•98522
•98039
•97560
•97087
•96628
•96153
•95690
•95238
94786
•94339
1
2
3
4
5
6
7
8
9
10
11
$
£ 31 2 55 I 2-86 -393157 -34965C| 95238 -943^^ l * ^
15* LOGARITHMIC* •
CoMSTUCTION OF T ABLE 8. &C.
The two first give the Amount, and Rebate, ot Presekt
Worth of 8 1, or £\ y from 1 to 3! years inclusively, at 5 and
6 pr. cent. Simple interest. They are calculated by making 1
dollar, or 1 pound the Principal in the first, and 1 dollar or
I pound the Amount in the second ; or dividing unity by the
srveral numbers in the first Table, gives the numbers in the
second Table. The third and fourth Tables are of the same
nature with the first two, and are therefore subject to the same
principle of construction.
Application and Use of the precbe&ing Tables.
CASE I.
To find the amount of any given turn/or year* and monthly
at 5) and b fir. cent, Simple Interest.
RULE.
To the Tabular number found in Table I, under the given
rate and opposite the time in years, add the number found on
the right of the decimal floint in, Table HI, under the given rate
and opposite the months, and multiply this sum by the princi-
pal ; and the product will be the amount sought.
EXAMPLES.
1. What will 8 100 amount to in 7 years and 8 'months, at .6
pr. cent pr. annum, Simple Interest ?
Tabular number = 1.42 = amount of gl for 7 years.
Do* .04a = do. for 8 months.
1.460
Multiply by .100 = principal,
8146.000= Ans.
2. Required the amount of 8318 50 cents* for 5 years, at 4
per cent per annum. Ans. 414 05cts.
3. What will 8753 215 cents amount to in 4 years and 7 months,!
at 5 per cent per annum ? Ans. 8925 86c* 3^m.
4. Required the amount of 1121. 10s. for 3 years and *
months, at 6 per cent. Ans. 1351. lis. 4d.
5. Required the amount of 1801. 8s. for 1 1 months, at 6 pei
cent per annum* Ans. 1901. 6s. 5d. 1 qr.
CASE II.
To find the Interest of any given sum for years and monthg^
at 5 and 6 fier cent*
RULE.
1 . Find the amount as before, from which subtract the priiuji*
palj and the remainder will be the interest.
ARITHMETICS. 121.
OR
2. Multiply the number found on the right of the decimal "
point in the table, by the principal, and the product is the inter-*
est required.
EXAMPLES.
1. What is the interest of $400 forbora 3^ years, at t pe>
cent per annum ?
Tabular number for 3 years = 1.20
Do. for 6 months. .02499
Multiply by
1.22499
400
Subtract
' 489.99600.
400.
Amount.
Principal.
Answer*
889.99600 =
= 890. fere.-
2. What will be the interest of 8210 35 cents, for 9£ years, at '
6 per cent per annum? Ans. 8123 05c. 4.75 m.*
3. What will be the interest on £45 10s. for 1 year and T
months, at 6 per cent ? V Ans. £4 6s. 5d. 1 qr.
4. Required the interest of £ 896 1 5s. for 7 months, at 5 per'
cent per annum. Ans. £27 2s. 1 id. 3 qrs.
CASE III.
Tojind the Rebate or Present worth of any given sum for
years and months.
RULE.
Multiply the Tabular number under the given rate and oppo-
site the time by the principal, and the product will be the fire-
sent worth.
EXAMPLES.
1. What is the rebate, or present worth of 8 100 due 1 year
hence, discounting at 6 per cent per annum ?
Tabular number = .943396
100
894.33.9/ T Ans*
2. Wfcat is the present worth of g 1 80 50 cents, due 5 years
hence, at 5 per cent per annum ? Ans. 8138 84c. 6m.
3 How much ready money will pay a debt of £ 1 12 10s. due
3 years hence, discounting at 6 per cent ? £95 6s. lid.
4. How much ready money is equal in value to £315 8s. duo
7 months hence, allowing 5 per cent discount ?
Ans,jr306 15s. 4d.
Note. When the discount is required, subtract the present
worth from the principal, and the remainder is the discount'.
i
lea TSOCARfTHMtCfc
5. What is the discount on£*$60, due 7 years hence, at 6 per
cent per annum? £Ans. 147 17s. ftd.
COxMPOUND INTEREST BY DECIMAL?.;
RULE.
i. Find the amount of gl, ov£\ y for one year at the given rate
per cent.
2. Involve the amount, thus found, to such a power, as is de-
noted by the number of years, and multiply this power by the
principal, or given sunt* and the product will be the amount re-
quired.
3. Subtract the principal from the amount, and the remainder
-will be the.itfcrfrf*
EXAMPLES.
I. What is jthe compound interest of £500 for 4 years, at 5
jper cent per annum ?
^The amount of 11. /or J year 7= 1.05 and
llW'X 5Q0>=? ^07.753125 = the amount.
/500
107.75312.5 *=,£107 15s. Ojd. interest required.
.2. What U the amount of #7>60 LOs. for 4 years, at 4 per ct. ?
Ans. 8891. 13s. 6id.
«f Table qf the amount of&\..or \ Lot 6 fier .cent per annum, for
months.
When the given time consists, of years and months, seek the
amount $f %\ &c. in the table for years, and the amount of g 1
Icc.in the foregoing table for the, months, and the continual pro-
duct of these tabular number* into vtfce principal,. will. give the
amount requiret 1 .
Note. Subtract the principal .from the amount, and the re»
mainder is. the compound interest?
ARITHMETICS,
1^?
1. Required the ahioirot of ^480 for 5 years land 6 mo0tKs,at
6 per cent per annum compound interest.
Tabular number of \L for 5 years = 1.334285
Do. " for 6 months = 1.02$56O
1.77*82931
480 = prinl,.
An*. /"66 1. 23*1 fee.
2. What will 8100 amount to, forborn 7 years and 10 months,
it 6 per cent pdr annum ? Ans. 8157 82c. 3m. '
3. What is the compound interest of 82 10 bQ cents, for 3
Jeers* at 6 per dent I Ans. 829 48c, 7m. +
4. What is the compound interest of 8QJ. 4s. for 9 yeara and
4 months, at 6 per cent per annum ? Ana. 471. 10s* Cf d.
Another method of computing Compound Interest for years^
months, and days. , -
RULE.
To the Logarithm of the principal, add the several logarithms
answering to the number of years, months, and days, found in
the following tables, and their sum will be the logarithm of the
amount required.
Lagarithmick Tabhesy at ^ jttr cent per annum, for years, months
and days*
iSvtatafc i din i Jit i aTu taTu I Ait 1 flat i?l~ iJiKi itf in lain 1 3*i i jtn> i jffin i il ■■ &
i F. J dec. /**». J F. I dec. fits. | Y. J dec. fits. \ V. | d&Jits.j:
U
" 3
If
( 6
ii r
II 8
i 1 o
(110
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i-OGARITHMICK
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1. Required the amount of 8436 50 cents, for 3 years, 8
months and 12 days.
Log. of principal = 2. 6399842
Add Log. of years = 0. 0759 1 80
Do. of months = 0. 0170330
Do. of days = o. 00085 70
Ans. &541 75 2. 7337922 Amount requirM
2. What will 8175 amount to, in lo years and 5 months, at 6
per cent per annum I Ans. 8321 28 cts.
CASE II.
The amount, rate, and time given, to find the firincifiqL
RULE.
T. Divide the amount of the given sum by the amount of £1,
or* £1, for the given time, and the quotient will be the principal ;
OR
*2. Multiply the present worth of £1, or £\, for the given
number of years, at the given rate, by the amount ; the product
will be the principal.
EXAMPLES.
1. What principal at 6 per cent per annum, will amount to
£757.4856, in 4 years ?
By Table I.
By Table II.
Multiply by the present worth > 757*4856
of 8 I for 4 years, at 6 per cent. 5 '792093
Ans. 8 600 principal.
2. What principal at 6 per cent, per annum, will amount tp
£757 9s. 8-Jd. in 4 years ? Ans. 600/.
ASITHMETIC&. 13&<
ARITHMETICAL PROGRESSION.
Any rank of numbers) increasing by a common excess, and
decreasing by a common difference, is said to be in Arithme-
tical Progression.
C 14.12.10.8.6 &c? r.
I 7. 6.5.4.3 kc. i DeteadmgieriM.
The numbers which form the series, are called the Terms of
the progression ; the first and last terms of which are called the
Extremes.
Any three of the five following terms being given, the other
two may be readily found.
1. The first term.
2. The last term.
3. The number of terms.
4. The common difference.
& The sum of all the terms.
PROBLEM I.
The first term, the last term, and the number of terms being
given, to find the sum of al) the terms.
RULE.
Multiply the sum of the extremes by the number of terms/
and half the product will be the answer.
EXAMPLES.
1. The first term of an arithmetical progression is 5, the last
term, 60, and the number of terms 12, required the sum of the
series. 5 -J- 60 X 12 -u 2 = 380 Ans.
2. The first term of an arithmetical progression is 3, the last
term 1 12, and the number of terms 18 ; required the sum of tho
series. Ans. 1035
3. How many strokes do the clocks of Venice, (which go to
24 o'clock,) strike in the compass of a day ? Ans. 3Q0
4. Suppose a man lay up 1 cent the first day of the year, 2
cents the second, and 3 the third day, and so on in arithmetical
progression, every day increasing 1 cent ; how much will he
have saved at the year's end ? Ans. 2667,95 cents.
5. A merchant bought 100 yards of cloth in arithmetical pro-
gression ; he gave 5 cents for the first yard, and 1 dollar for.
the last, what did the cloth amount to ? Ans. g52,50 cts.
6. If 100 stones be placed in a right line, exactly a yard asun-
der, and the first a yard from a basket, what length of ground
will that man go, who gathers them up singly, returning with
them one by one to the basket ?
Ans. 5 miles, 233 rods,. 2 yards*
17
l& > «>GAR*THMICk
PROBLEM II.
The first term, the last term* and the number of tetms giy£n>
to find the Common Difference. '■
RULE.
Divide the difference of the extremes by the number of term*
less by i, and the quotient will be the Common difference re-
quired.
EXAMPLES.
1. If the extremes be 3 and 19, and the number of terms 9,
what is the common difference, and the sum of the whole series.?
19
Extremes J ! J J[
— 22
9 — l = 8)16(2 common difference 9
16 - —
— i) 198(99 sum of series.
2. A man had 10 sons, whose several ages differed alike, thA
youngest was 4 years old, and the oldest 40 ; what was the
common difference of their ages ? Ans. 4 years.
3. A man travels from Manchester to London in 6 days;
every days journey was greater than the preceeding one, by a
common excess ; he traveled 20 miles the first day, and 40 miles
the last ; what was the common increase of each succeeding
day's, journey and the distance from Manchester to London ?
Ans. daily increase 4 miles, and the distance of journey
180 miles.
PROBLEM III.
The two extremes and the common difference given, to find
the number of terms.
RULE.
Divide the difference of the extremes by the common differ-
ence, and the quotient, increased by 1, is the number of terms
Required.
EXAMPLES.
1. If the extremes be 3 and 19, and the common differejicej2>
what is the number of terms ?
19 — 3-5-2 = 8+1 = 9 Ans.
2. A man, going a journey, travelled the first day 3 miles
and the last day 43, and increased his journey every day 5 miles ;
how many days did he travel ? Ans. 9 days.
ARITHMETIC^. • ,1,3?
GEOMETRICAL PROGRESSION.
Any series of numbers are in Geometrical Progression*
.when the several terms increase bya coinmon multiplier, or de-
crease by a common divisor ; — Thus, 3, 6, 12, 24, 48 &c~ is a
series in geometrioal progression, increasing by the common.
Multiplier 2 ; and 81, 27, 9, 3, 1, 8cc. is a series in geometrical,
progression, decreasing by the common divisor 3.
The number, by which the series is constantly increased, or
diminished, is called the Ratio.
PROBLEM I.
Given the first term, the last term, (or extremes) and the ra**
tio, to find the sum of the series.
RULE.
Multiply the last term by the ratio, and from the product sub-
tract the first term, and the rem ainder, divided by the ratio less
1, will give the sum of all the terms of the series.
examples.
1. The first term of a series in geometrical progression is o«.
the last term 531441, and the ratio 3 ; required the sum of all
the terms.
The series is 3,9,27,8 1,243,729,2 187,6561, 19683,5904$,
177147,531441. Then,
531441 X 3)— 3(-~3 — 1 c= 797160 Ans.
2. The extremes of a geometrical progression are 1 and '
65536, and the ratio 4 j what is the sum of the series ?
Ans. 87381
PROBLEM II.
Given th,e first term and the ratio, to find any other term as-
signed,
CASE I.
When thcjirst term of the series and the ratio are equal,*
1. Write down a few of the leading terms of the series, and
place their indices over them, beginning with an unit, or 1.
2. Add together such of the most convenient indices, as wilL
make up the entire index to the sum required.
* Note. When the first term of the series is equal to the ra«
tio, the indices must begin with an unit, and the indices added -
roust make the entire index of the term required ; but if
the first term be greater, or less than the ratio, the indices must
begin with a cypher, and the indices added must make an index
less by I than the number, expressing the plaqp of the t^rni
it* LOGARITHMIC*
3. Multiply the terms of the geometrical series belonging t^
those indices, continually together, and the product will be the
term sought.
EXAMPLES
•If the first term be 2 and the ratio 2, what is the Istb
ftrm I
ThtM 5 l>2 >*> 4 » 5l 6 Indices.
in ' {2,4,8,16,32,64 Leading terms.
Then 4+S T 6a= 15 the index of 15th. term, and product
df 16, X 32 X 64, =• 32768 term required.
% A merchant bought 22 hhd.of wine for 2 mills for the first
&hd. 4 for the second, 8 for the third, and soon in duplicate pro-
portion geometrically ; what did the whole amount to at that
rate I
The 2 2d. or last term is 4194304.
v Then 2X4194304—2
■ mi =. 8398606, the sum of all the terms
2—1
(by PaoB. I.) = $ 8398,60t. 6m. Ans.
3. A labourer agreed to work one whole year for a rich mi-
dfer, to receive no other reward than 3 farthings for the 1st*
month, 2£d. for the 2d. month, 6f d. for the 3d. month, and so
on, in triple proportion geometrically ; what did his wages
amount to in one year, and what was the average price of each
day's labour I Ans. His wages for one year amounted to
£"850 7s. 6d. and the average price of each day was £ 2 7s. 6d.
CASE II.
When the first term of the series and the ratio aTe different*
that is, when the first term is either greater or less than thfe
ratio.
RULE.
1. Write a few of the leading terms as before, and begin
their indices with a cypher.
2.' Add together the most convenient indices to make an in
dex, less by 1 than the number expressing the term sought.
3. Multiply the terms of the geometrical series together
belonging to those indices, and make the product a dividend.
4. Raise the first term to a power whose index is one leas
than the number of terms multiplied, and make the result a di-
visor.
5. Divide the dividend by the divisor, and the quotient will
be the term sought.
•» » - ■ ■
* Note. See this principle explained by logarithms,— page 44
ARITHMETICS. *»
EXAMPLES. ,
1. If the first term of a geometrical series be S, and the nm
3, what is the 1 1th term ?
Thn* S°> *> 3 > 3 » 4 > Indices.
Uf {5,15,45,135,405, Leading terms.
And 1+2 + 3+4 ae 10 the index of the 11th. term.
15X45X135X405 = 283943125
=2263545 the 1 ltlxc
53,-. 125 [term required.
Here the number of terms multipled are 4 ; therefor© the*
Jst. term raised to the 4th. power less by 1 is the 3d. power, or
cube of 5 = 125 the divisor.
2. What debt can be discharged in a year, by paying 2 cents
for the first month, 8 cents for the second, 32 cents for the
third month, and so on in quadruple proportion, for each
month ? Ans. 8 111848 10 cts.
3. An ignorant horse jockey being employed to purchase a
number of horses for shipping, very readily agreed with a gen-
tleman, well skilled in numbers, for 28, upon condition that he
should give 1 cent for the first horse, 5 for the second, 25 foT
the third horse, and so on in quintuple proportion to the last
horse ; what did they come to at that rate, and how much did
they cost per head ?
Ans. the horses came to g 307708728652954101,56 cts. and
the average price was, $ 10989597451855503,62 cts. 7Jm. per
licad.
4. What will a horse cost, computing his worth in geometri-
cal progression by the nails in his shoes, at a farthing for the
first nail, 3 farthings for the second, and so on in triple propor-
tion to the last, or 32d. nail ? Ans. £9651 14681693 13s. 4d.
5. A young man skilled in numbers, agreed with a farmer to
work for him 1 1 years, without any other reward than the pro-
duce of one wheat corn for the first year, and that produce to
be sowed the second year, and so on from year to year, till the
end of the time, allowing the increase to be in a tenfold propor-
tion ; what quantity of wheat is due for such service, and to
what does it amount, at 8,150 per bushel ?
Ans. w 226056! bushels, allowing 768 whqat corns to make a
pint; and the amount is £339084 18c. 24m.
6. What will be the value of a 64 gun-ship, reckoning 1 pen-
ny for the fir6t gun, 2 pence for the second, 4 pence for the
third gun, and so on to the Jast, in duplicate proportion ?
Ans. £786 1435640456465 IS. i d.
130 LOGARITHMICK
7. Suppose America should agree to build 144 ships of the
Jthe for Great Britain, at the rate of but 1 farthing for the first
ship, 2 for the second, 4 for the third, and so to increase, in a
duplicate proportion to the last ; what would they all amount to
at that rate, and how many globes of standard gold, equal in
magnitude to the earth we inhabit, could be formed from the
mass, allowing a cubic inch of gold to be worth £53 2s 8d. ?
1 the 9th Ship=: 256 the 18th Ship = 131072
131072
262144
9175,04
131072
393216
65536 131073
131072 ,' ■ ■/ ■
17179869184
The 36 Ship 34359738368
34359738368
274877906944
206158430208
103079215104
2748779Q6944
103079215104
240518168$T6
309237645312.
* 171798691840
103079215104
137438953472
103079215104
■ ■ ' ' ■ > ■
1180591620717*11303424
The72d. Ships 2361183241434822606848
As multiplying the index of any term in a geometrick series*,
by a given number, gives the index for that fiower of the term,
denoted by the multiplying number ; therefore, as 72 X 2 = 144,
raise the 72d term to the second power, that is, multiply it by
itself, and that product by the ratio, the last product will be the
amount of the last term, which by (PROB. 1 .) will give the sum
of all the series : as follows.
4BIJHMETICX *M
Tf)fe72d. Ship a 2361183241434822606343
2361183241434822606848
18889465931478580854784
9444732965739290427392
18889465931478580854784
14167099448608935641088
141670994486089356410880
4722366482869645213696
47223664828696452 1 5696
1 888946593 1 478580854784
94447329657,59290427392
7083549724304467820544
444732965739290427392 *
2361 183241434822606848
9444732965739290427392
4722366482869645213696
7083549724304467820544
188.89465931478580854784
. 236I18324143482260G848
2561183241434822606848
14f67p99448608935641088
. 7083549724304467820544
-4722366482869645213696
— m I"
5575186299632655785383929568162090376495104 Ship.
1 1 15037259926531 1570T67859 136324180752990208 ss 144
4)22300745198530623141535718272648361505980415 T. Sum.
12)5575186299632655785383929568162090376495103
30)464597191636054648781994130680174198041258— 7d.
£232298595818O27324390997O6.53400870?9O2O12 _ 18s. 7d.
Which is the exact amount of the whole number of ships ; and
ia»computed according to the conditions of the question.
Now to reduce this to solid gold, divide the amount by 53J. :
2s. 8d. (or 12752d.) and it will give
,4372004174743212871580376161734358302 the number of solid
inches the mass would contain. And in order to compare {his
«mass with the solidity of the earth, this also must be reduced to
inches. The number of solid miles contained in thfc*eartlr(see
page 99) is 263858 1 49 1 20, which multiplied by 254358&8873feoOO
the number of cubic inches in 1 mile, gives
66646680917786616312320000 solid inches $ by whicfe 'divide
the number of sofid inches in the whole mass, and the quotient
will be the number of globes, (equal in magnitude to the earth,)
' contained In the mass, which is the Answer required s as follows.
_132. LOGARITHMICK
SOU. "««*J SOXIB INCHES IN THE WJIOtE MASS. 5**™"* «
ip THB KARTH. ) £ GLOBES.
666466809 177 8-) Ans.
661631232QOOQ)4372004l7474321287158037616l734S583Q2(655997285ir
S9988008550671969787392000Q
373203319676015892841176161
333233404588933081561600000
399699150870828112795761617
333233404588933081561600000
6646574628 1 89503 1 234 1 6 1 6 1 73
599820128260079546810880000
648373345588707755307361734
59982012826007954681088000O
485532 1 732862820849648 1 7343
466526766424506314I8624000O
19OO54O68516757707795773435
133293361835573232624640000
567607067811844751711334358
53317344734229293049856000Q
344736204695518212127743583
3332334045889330ai 56 1600000
t
115028001065851305661435830
66646680917786616312320000
483813201480646893491158302
• 466526766424506314186240000
_^_ *
And something over.
Tims we have proved, what at first might not seem to be easy of be-
lief, that the whole amount in Sterling Money is twenty three thousand
two hundred and twenty nine miliions of millions of millions of millions of
millions of millions, eight hundred fifty nine thousand five hundred and
eighty one millions of millions of millions of millions of miliions, eight-
hundred two thousand seven hundred and thirty two millions of millions
of millions of millions, four hundred thirty nine thousand and ninety nine
millions of millions of miliions, seven hundred six thousand five hundred
and thirty four millions of millions, eighty seven thousand and ninct£.nine
millions, nine hundred two thousand and twelve pounds, eighteen i&H-
Tings and seven pence* * V
ARITHMETIC^.
And the number of solid globes of gold, equal to tke earth we
inhabit, is sixty live thousand five hundred and ninety nine mil-
lions, seven hundred twenty eight thousand five hundred and
seventeqp.
POSITION.
Position is a rule, which by false, or supposed numbers tak-
en at pleasure, discovers the true one required.
It is divided into two parts, Single and Double.
SINGLE POSITION.
Single Position teaches to resolve such questions, whose re-
sults are proportional to their supposition ; and is when the
proportions of the required number are implied in the conditions
of the question,
RULE.
1. Take any number and perform the same operations with
it, as are described to be performed in the question.
2. Then say ; as the result of the operation ; is to the given
sum in the question : : so is the supposed number : to the true
one required.
examples;
1. A Schoolmaster being asked how many sc hollars he had,
said, if I had as many more as I now have, half as many, and one
fourth as many, I should then have 99 ; how many scholiars
had he ? As 1 10 : 99 : : 40 ; 36 Ans.
Suppose he had 40 36
as many =40 18
\ as many = 20 9
£ as many =10 —
— Proof 99
Result 110
Or,— As 110: 40: : 99 : 36 the Ans.
2. A person after spending one third and one fourth of hi^s
money* had £ 60 left ; what had he at first ?* N Ans. g 144.
3. What number is that, a sixth part of which exceeds an
eighth part of it by 20 ? Ans. 480
4. What sum of money is that, whose third part, fourth part,
and fifth part, added together, amount to 94 dollars ?
Ans. g 120 '
5. In a mixture of corn and oats, £ of the whole plus 25 bush-
"* els was corn, -J. part minus 5 bushels was oats ; how many bush-
els were there of each ? Ans. 85 of corn, and 35 of oats.
18
\
r 7
LOGARITHMICK
6. What number is that, from which if 5 be subtracted, f of
the remainder will be 40 ? Ans. 65.
7. Two travellers, A. and B. 360 miles apart, travel toward*
each other tilflhey meet. A's progress is 10 miles in an hour,
and B's 8 ; how far does each travel before they meet ?
Ans. A. goes 200 miles, and 13. 160.
8. If a certain number be divided by 12, the quotient, dividend
and divisor added together will amount to 64 ; wBat is the num-
ber ? Ans 64.
9. A. man spent one tfcircl of hi$ Bfe fa England, one fourth
of it in Scotland, and the remainder of it, which was 20 years*
in the United States ; to what age did he live ?
Ans. to the age of 4$:
DOUBLE POSITION.
Double Position teaches to resolve questions' by making
two suppositions of false numbers.
RULE.
1. Take any two convenient numbers, and proceed with each
according to the conditions of the question.
2. Find how much the results are different from the result in
the question.
3. Multiply the first position by the last errour, and the la^t
position by the first errour.
4. If the errours are alike, divide the difference of the pro-
ducts by the difference of the errours, and the quotient will be
the answer.
5. If the errours are unlike, divide the sum of the products
by the sum of the errours, and the quotient will be the answer.
Note. The errours are said to be alike, when they are both too
great, or both to© small ; and unlike, when one is too great ami
the other too small.
ARITHMETICS IS*
EXAMPLES.
1. The ages of 4 persons amount together, to 109 years, A i*
7 years older than B, and C is 10 years younger than A, and D
;h | as old as A ; required the age of each.
1st. Suppose A's age =a 40 ad. Suppose A. s= 30
B's " = 33 B. = 23
C's « =t 50 C> = 20
D's « = 24 D. = 18
127 — 9i
—109 109
1st. errour 18 2d. errour 18
The errours being unlike or one too great, and the other too
small.
Pos. Err.
40 30
Therefore
X
**
AV » 35
B J s s= 28
C's = 25
18 18 (JO'S = 21
30 40
— Proof 109
540 720
18+ 18 =* 36)1260(35 = A*s agft.
2. Three merchants enter into partnership with a stock of
$ 1 140, A put in a certain sum, B put in one third as much as
A and 8 50 more, and C put in twice as much as B, together
with a fifth of what A put in ; what was each one's respective
•hare in the stock !
Ans, A put in g 450, B g 200, and C S 490
3. The ages of two persons A and B are such, that 7 veapi
ago, A was three times as old as B ; and 7 years hence, A will
be twice as old as B ; what are their respective ages I
An«. A 'sage is 49, knd B's21 years
4. Three persons A, B, and C, purchase ahorse for 100 dol-
lars, but neither is able to pay for the whole : the payment
would require
The whole of A's money, together with half of B's ; or
The, whole of B's, with one third of C's ; or
The whale of C's, with one fourth of A's ;
How much money bad each ?
Ans. A, had 8 64., B g 72, and C g 84
5. Tire sum of the distances which 3 persons travelled, is 62
nujes ; A travelled 4 tildes as^ far as. C, added to twice thedia*
136 LOGARITHMICK
fence that fi travelled, and had C travelled IT times as far as he
did, he would then have travelled 3 times as far as B, added to
twice the distance that A travelled ; required their respective
distances ? Ans. A travelled 46 miles, B 9, and C 7 miles.
PERMUTATION OF QUANTITIES.
The permutation or variation of quantities is the
showing how many different ways the order or position of any
given number of things may be change^.
To find the number of permutations or changes, that can be
made of any given number of things, all different from one
another.
RULE.
Multiply all the terms of the natural series of numbers, from
one up to the given number, continually together, and the last
product will be the answer required.
EXAMPLES.
1. How many changes can be made of the fl
fetters in the word and ?
Proof
\
and
a d n
n a d
n d a
dan
d n a
I X 2 X 3 =- 6 Ans. j 5
i«
2. How many changes can be rung on 12 bells ?
Ans. 479001600
S. How long could a family of 9 persons vary their position.
c£ dinner ? Ans. 994 years 80 days.
4. How many changes can be made (in position) of the 8
notes in musick ? Ans. 40320
5. How many variations may be made of the letters in the
English alphabet ? Ans. 403291461 *2 66056355 8 4000000
CONSTRUCTION OF THE FOLLOWING TABLES
BELONGING TO COMPOUND INTEREST-
The construction of these Tables by logarithms, will
be best understood by the following proposition. Viz.
Between two numbers given, to find any number t>f mean
proportionals required.
ARITHMETICS I3t
RULE.
1. From the logarithm of the greater number subtract the
logarithm of the less, and divide the remainder by the number,
of means increased by 1
2. Add the quotient to the logarithm of the less, number, and
the sum will be the logarithm of the 1st mean proportional re-
quired.
3. To the logarithm last found, add the said quotient, and the
sum will be the logarithm of the second mean proportional ;
and thus proceed, always adding the said quotient to the loga-,
rithm of the last proportional found, as far as the question re*
quires.
examples.
Required to find, between 16 and 64, 5 xpean proportionals.
Logarithm of 64 = 1-8061800
Do. #f 16 = l-204i 200
The difference ±= 06020600
To \ part for 5 means = 0-1003433
Add logarithm of 1 6 1 -204 1 200 ,
1 st. Mean proportional = 20- 1 58 = 1*3044633
To which add said*quotient = ' 0-1003433
2d. Mean proportional =* 25-398 «= 1 -4048066
Add quotient 0-1003433
3d. Mean proportional =32 = 1-5051499
Add quotient 0*1003433
4th. Mean proportional = 40-3 1 7 = 1 .6054932
Add quotient 0-1003433
5th Mean proportional = 50*796 =• 1.7058365
For the construction of the first two following tables are sev-
eral methods used. We shall mention only that which is irtost
easy and expeditious ; which is by logarithms.
For the first Table thus : Find the amount (as already
taught) of 1 dollar or 1 pound for 40 years at the given rate per
cent, and between the logarithm of the amount and the loga-
rithm of the rate find 40 geometrical mean proportionals, by the
last proposition ; and these will be the logarithms of the num-
bers in the first Table? Or,
i
**• LOGARITHMICK
If we add the logarithm of the rate continually to itself, it wilt
give the same result ; Thus, adding the logarithm of the rate
to itself, gives the logarithm belonging to the second year,
*nd to this sum adding again the logarithm of the rate, gives
the rogarithm of the number belonging to the third year, &c.
Or, if ^>u multiply the logarithm of the rate, by the numbers
1, 2,3, 4, 5, 6, &c. gives the logarithm of the numbers belong-
ing to those respective years ;
' And for the numbers in the second Table find the Arithme-
tical Complements of the logarithms of the numbers in the first
Table, and you will have the logarithms of the numbers in the
second Table.
The Logarithmical differences of 045, or 05, or 06,(being the
rates here used, minus unity,) and the numbers in Table I, are
the logarithms of the numbers in Table III.
If, from the logarithms of the numbers in Table III. you
subtract the logarithms of the numbers in Table I. you will
bave the logarithms of the numbers in Table IV.
And their Arithmetical Complements are the logarithms of
th* numbers in fable V.
ARITHMETIC^
l J*
IfABLK I. Shewing the amount
of% l,or£\frpm 1 year to 40.
TABtE II. Shewing the
present value of% 1, or £i
due at the end of any num-
ber of years, from \ to 40
Y.
4f fir. ct.\5 fir. ct.
6 fir. cs.
4\ fir. ct
Sfir.m.
f>fir. ct.
\
L 045000
1-050000
'1*060000
•956931
J5I4381
943396
2
t 092025
1- 102500
1- 123600
•915730
907030
•88999J&
S
M41166
1-157625
1-191016
•876297
863838
•839619
4
1-192518
1*215506
1-262476
•838561
822702
•792093
5
■
6
1*246181
1-276281
1-338225
•802451
783526
•747258
1.302260
1-340095
1*418519
•767896
•746215
•704960
7
1*360861
1-407100
1*503630
•73482fc
710681
•665057
6
1-422100
1-477455
1*593848
•703185
676839
•627412
9
1*486095
1-551328
1-689478
•672904
6446C9
•591898
10
11
1*552969
1-628894
1-790847
•643928
613913
•558394
k
1-622853
1-710339
1-898298
•616199
^584679
•562787
12
1-695881
1-795856
2012196
•589664
■556837
•496969
13
1-77219*
1*885649
2132928
•564271
53oS21
•468839
N
14
1-851944
I -979931
2*260903
•539973
•505068
•442300
15
1-935282
2078928
2*396558
•516720
•494469
•481017
•417265
16
2*022370
2* 182874
2-547271
•458311
•393647
17
2- 11 3376 2-2920 18
2-692772
•473176
•436297
•371364
18
2*208478 2*406619
2-854339
•452800
415521
350343
19
2*307860|2-526930
3025599
! -433302
•395734
•330513
20
2-411714
2-653297
2-785962
3-207135
•414643
•396787
•376889-311804
21
■ ■
2-520241
3*399563
•358942/294155
22
2*633652 ! 2*92526Q
3.603537
•379701
.341850
•277505
23
2-752166 3-071523
3.819749
•363350
•325571
•261797
24
2-876013>225099
4.048934
•347703
310068
•246978
25
3-005434
3 386354
3-555672
4,291870
•332731
305303
•232998
26
3-140679
4.549382
•318402
281241
•219810
27
3-282009;3 733456
4.822345
•504691
267848
•207368
28
3-429699 3-920129
5.111686
•291571
255094
175630
29
3-584036 4-11 61 $5
5.418387
•279015
•242946
184556
30
3-7453 18'4-32 1942
5.743491
•267000
231377
17411o
31
3-913857J4-538039
6.088 100
•255552
220359
164255
32
408998 1;4-7 64941
6.453386
•244500
209866
154957
3.3
4-274030 5003188
6.840589
•233971
199872
146186
o4
4-466361
5253347
7.251025
•223896
190355
137912
35
4-667347
5-516015
7.686086
•214251
•205028
•181290
130105
36
4*877378
5 791810
8.147252
172057
122741
57
,5-096860
6 081406 1 8.636067
•196299
164436
115793
38
5326219
6385477J 9.154252
•187750
•156eo")
109182
39
$565899
6 704751? 9 703507
•179659
149148
103002
40
5-816464
7 039988 10285717
•i7i9M-VAa^-vV»yivt^\
ho
LQGARITHMICK
Ta*l* III. Shotting the amount
of$\ or £1 annuity for any num-
ber of 'year *, from 1 to 40.
Table IV^ Shewing tht
present worth of $ 1 or £1
annuity , for any number of
y ear H y from 1 to 40.
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
ZB
29
50
31
32
33
34
35
56
>7
38
39
1 000000
2*045000
3-137025
4-278191
5-470710
6716892
8-019152
9-380014
10-802114
12-288200
1 3.84 1179
15.494032
17-159913
18.932109
20-784054
22-719337
24741707
26-855084
29-063562
31-371423
33:783137
36-303578
38-937030
41-689196
44.565210
47-570645
50*711324
53-993333
57-423Q33
61-00,7067
64-752388
68-666245
72-756226
77-030256
81-496618
1 -000000
2-050000
3-152505
4-310125
5-525631
6-801913
8- 142008
9549109
11026564
12-577892
14-206787
15917126
17-712985
19-598632
21-578563
23-667492
25-840366
28-132385
30-529003
33-065954
35-719252
38-505214
41-430475
44-501999
47-727099
51*114454
54-669126
58-402583
62-32271
66-438847
70-760790
75-298829
80-063771
85066959
90-320307
£6-16396f
9 1 04 1 344
96-138206
101-464424
95-836323
101-628139
107-709546
114095025
120-799774,
6 fir, cent
1 000000
2 O60000
31836oo
4-374916
5 637093
6-975318
8-393837
9-897467
11-491315
13-180794
14-971642
16-869940
18-882132
2 10 1 5064
23-275968
25-672527
28-212879
36-905652
33-759991
36-785590
39-992725
43-392289
46-995826
50-815576
54-864510
59-156381
63705763
68-528109
73-639796
79-058183
84801674
90.889775
97-343161
104183751
111-434776
119' 120863
127-268114
435-904201
145*058453
15476196\
^\h r. cf 5 fir. cl. 6 fir. ct.
0-95694
1-87267
2-74896
3-58752
438997
515789
5-89270
6.59589
7-26879
7-9 1272
0-95238
I 85941
2-72325
3-54595
4-32948
5-07569
5-78637
6-46321
7-10782
7-72x73
8-52892- 8-30641
9-11858? 8-86325
9-68285= 9-39357
10-22282;. 9-89864
10-73954 10-37966
11-2340M0-85777
1 1-707 19 ! II'27407
12-l5099|lI-68958
12-59329 12-08532
1300793 l2-4622i
13-40793 I2-82I 15
13-78442 1 3- 1 63oo
14 14777
14-49548
I48282i
15-14661
15-45150
15 74287
1 6-02189
16 28889
16-54439
16 78889
1702286
17-24676
17-46I0I
17-66604
17-86224
18 04999
18-22965
\8-40|58
13-48807
13-79864
1409394
14 37518
l4-643o3
1 4-898 1 3
I*-x4lo7
15-37245
I55928I
l5-8o268
l6oo255
16 i929o
16-37419
0-94339
1.83339
2-6730X
3-465IO
4-2x236
4-91732
5-58238
6-2o976
6-8ol69
7-360O8
7*88687
838384
8<85268
9-29498
9-71225
IO 10589
10-47726
10 827fo
IIl58ii
1 1 46992
11-7640?
12-04X58
l~-3o338
l2-55o35
12*78335
i3oo3i6
l3-2ro53
13-40616
l3-59o7
13-76483
13 92908
I408398
14-22917
44 366 13
14 49533 1
16-54685
167x129
16-86789
i7oi7o4
l2.i59o9
1461722
14 7S211
1 4 84048 1
I4P4270J
15-Q5913*
ARITHMETICS
141
The annuity which 8 1, or £ 1 will purchase for any number. of
years to come y from 1 to 40.
• y"»»g» qeaaaaa « aaaa»o»tteaa«aaaa«a » B aaa ae e eaea»
* * - **' Ai fler cent ' 5 fi er ccnt - I 6 fi*r cent. I TrTf
^ 4 1 04500 105000 I 1-06000 1 •
•53400
•36377
•27874
•22779
•19388
•16970
•15161
•13757
•12638
•10967
•10327
•09782
•093 1 1
•08901
•08542
•08224
•07941,
'07688
•07460
•07254
•07068
'06899
"06744
•06602
•06472
•06352
•06241
•06139
•06044
•0595.6
•05874
•05798
•05727
•05660
•05224
•05540
•0$485
•05434
53780
•36721
28201
2309]T
19702
17282
15473
14069
12950
•12039,
•11282
•10645
•10102
•09624
•09227
'08870
•08555
•08274
•08024
«— -*-i--r i ».|
•07800
•07597
•07414
•07247
•07095
•06956
•06829
•06712
•06604
•06505
•06413
•06328
•06249
•06175
«O6107
•06043
•05984
•05928
•05876
•05?28
•54544
•37411
•28859
•23739
•20336*
•17913
•16103
•14702
•13587
•09895
16
•09544
17
•09235
18
•08962
19
•08718
20
•08500
21
•08303
•08128
•07968
•07823?
•07690
•07570
•07459
•07358
•07272
07179
<07100
•07027
•06959
•06899
•06839
•06785
•06735
<Q6689
•06646
6
7
8
9
10
11
12
13
14
15
V
\
K
\
V
s
V
V
V
I
\
PRACTICAL ASTRONOMY.
Containing a number of Astronomical Tables, and are an easy
method of calculating the times of New and Full Moons, and
Eclipses by them.
Of Astronomical Tables and their Construction.
IN constructing tables for computing, at any given instant,
the places of the Sun, Moon, and Planets, the first step is to de»
termine, from a series of accurate observations, the time in
which those bodies describe a space of 360 degrees, or perform
a complete revolution round the Sun, or primary Planet.
When this important element is exactly ascertained, we can
easily find, by simple proportion, the space which any Planet de-
scribes in any number of years, months, days, hours, minutes,
and seconds, upon the supposition that it moves uniformly, or
describes equal spaces in equal time, in the circumference of a
circle.
But as it has been found from a long series of observations,
that all the bodies of the solar system, move in eliptical orbits
round the Sun; or their primary Planet, placed in one of the fo-
ci, we must next determine the form of their orbits, or the na-
ture of the ellipse.* which they describe.
The diameters of the Sun and Moon therefore, subtend differ-
ent angles at different times, as they aije nearer, or more remote
from the observer's eye. This proves that the Sun; and Moon
are constantly changipg their distances from the Earth,; and
they are once at their greatest, and once at their least distance
from it, in little more than a complete revolution.
The gradual differences of these angles are not what they
would be, if the luminaries moved in circular orbits, the Earth
* An eiiipse is a curvilinear figure of an oblate oval form,
having two centres called Foci, ortfoucuses: The Sun is in
the focus of the Earths orbit, and the Earth is in or near that of
the Moon's orbit. ' '
ARITHMETICS 1 43
being supposed to be placed at some distance from the centre
of the orbit, and the centre of the Earth to be in the lower focus
of each orbit.
The fartherest point of each orbit from the Earth's centre is
called the Apogee, and the nearest point is called the Perigee*
These points are diametrically opposite to each other.
Astronomers divide each orbit into 12 equal parts called
Signs ; each sign into 30 equal parts called Degrees ; each
degree into 60 equal parts called Minuets ; each minuet into
60 equal parts called Seconds.
The distance of the Sun or Moon from any given point of its
orbit, is reckoned in signs, degrees, minuets, and seconds. Here*
in is meant the distance that the luminary has moved through
from any given point : and not the space it is short of it in com*
ing round again, though it be ever so little.
The distance of the Sun or Moon from its apogee at any given
time, is called its Mean Anomaly : therefore when the body is
in its afiogccy its anomaly is 0, and in its ficrigee, it is 6 signs.
The motion of the Sun and Moon are observed to be contin-
ually accellerated from their apogee to their perigee, and afc
gradually retarded from their perigee to their apogee ; moving
with the greatest velocity when the anomaly is 0, and with the
least, when the anomaly is 6 signs.
When the luminary is in its apogee or its perigee, its place
is the same as it would be, if its motion were equable in all parts,
of its orbit. The supposed equable motions are called Mean ;
the unequable motions are justly called the True.
The mean place of the Sun or Moon is always forwarder than
the true place,* while the luminary is moving from its apogee
to its perigee ; and the true place is always forwarder than the
mean, while the luminary is moving from its perigee to its apo-
gee. In the former case the anomaly is always less than 6 signs;
and in Jhe latter case, more.
The Moon's orbit crosses the ecliptick in two opposite points,
which are called her Nodes ; and the time in which she revolves
from the Sun to the Sun again, (or from change to change) is
called a Lunation, and would always consist of 29 days, lfc
hours, 44 minuets, 3 seconds, 2 thirds, and 58 fourths, if the mo-
tions of the Sun and Moon were always equable. Hence, 1 2 mean
lunations contain 354 days, 8 hours, 48 minuets, 36 seconds, 3$
thirds, and 40 fourths, which is 10 days, 21 hours, 11 minuets*
23 seconds, 24 thirds, and 20 fourths, less than the length of a
common Julian Year, consisting of 365 days 6 hours; and IS
*The point of the ecliptick in which the Sun or Moon isittL
any moment of time is called the place of the Sua or Moonai'
that time,
144 LOGARITHMICK
mean lunations contain 383 days, 21 hours, 32 minuets, 59 sec-
onds, 38 thirds, and 38 fourths, which exceeds the length of a
common Julian Year, by 18 days, 15 hours, 32 minuets, 39
seconds, 38 thirds, and 38 fourths.
The mean time of New Moon being found for any given year
and month, as suppose for March 1850, New Style, if this mean
New Moon happens later than the 1 1th day of March, then 12
mean lunations, added to the time of this mean New Moon, will
give the time of the mean New Moon in March 1851, after aba-
ting 365 days. But when the mean New Moon happens to be
before the 1 1th of March, we must add 13 mean lunations, in
order to have the time of mean New Moon in March the year
following ; observing always to subtract 365 days in common
years, and 366 days in leap-years, from the sum of this addition.
Thus, A. D. 1850, JWw Style, the time of mean New Moon
in March-, was the 12th day, at 22 hours and 1 1 seconds past the
noon of that day (viz. at 1 1 seconds past X in the morning of the
J 3th day, according to common reckoning.) To this we must
add 12 mean luxations, or 354 days, 8 hours, 48 minuets, 36
seconds, 35 thirds, and 40 fourths, and the sum will be 367 days,
6 hours, 48 minuets, 47 seconds, 35 thirds and 40 fourths ; from
which subtract 365 days, because the year 1851 is a common
year, and there will remain 2 days, 6 hours, 48 minuets? 47
seconds', 35 thirds and 40 fourths, for the time of mean New-
Moon in March, A. D. 1851.
Now to find the mean time of New Moon in March A. D.
1852, we must add 13 mean lunations to the mean time of New
Moon in the next precfecding year, (because it happened before
the 1 1th day)a'nd the sum will be 386 days, 4 hours 21 minuets 27
seconds 13 thirds and 18 fourths ; from which subtract 366 days,
because the year 1852 is a leap-year, and there will remain 20
days 4 hours 21 minuets 27 seconds 13 thirds and 18 fourths, to
be set down for the time of mean New Moon, in March, A; D..
1852
In this manner was the first two of the fallowing tables con-
structed to seconds, thirds, fourths ; and then written out to
the nearest second. The reason why Astronomers choose to be-
gin the year with March, is to avoid the inconvenience of ad-
ding a day to the tabular time in leap-years after February, o*
subtracting a day therefrom in January and February in those
years ; to which all tables of this kind are subject, which be-
gin tho year with January, in calculating the times of New or
Full Moons.
The mean anomalies of the Sun and Moon, and the Sun's
mean distance from the ascending node of the Moon's orbit, are
set down in Table III, from one to 1 3 mean lunations.
The numbers, for 12 lunations, being added to the radical
anomalies of the Sun and Moon, and to the Sun's mean distance
ARITHMETICS U*
from the Moon's ascending node, at the mean time of New
Moon in March 1850, (Table II.) will give their mean anoma-
lies, and the Sun's mean distance from the node, at the time of
New Moon in March 1851 ; and being added for 13 lunations
to those for 1851, will give them for the time of mean New
Moon in March 1852. And so on as far^as you please to con-
tinue the table, (which is here carried on from 1752, to the year
1900,) always rejecting 12 signs when their sum exceeds J 2,
and setting down the remainder as the proper quantity.
If the number of years belonging to A. D. 1700 (in Table I.)
be subtracted from those belonging to 1800, we shall have their
whole differences in 100 complete Julian years ; which accord-
ingly we find to be 4 days 8 hours 10 minuets 52 seconds IS
thirds 40 fourths, with respect to the time of mean New Moon.
These being added together 60 times, (always taking care to
throw off a whole lunation when the days exceed 29*) making
up 60 centuries, or 6000 years, as in Table VI. whicu was car-
ried on to seconds, thirds, ami fourths ; and then written out to
the nearest second. In the same manner were the respective
anomalies and the Sun's distance from the node found, for these
centural years ;and then (for want of room) written out only to
the nearest minuet, which is sufficient in whole centuries. By
means of these two tables, we may find the time of any mean
New Moon in March, together with the anomalies of the Sun
and Moon, and the Sun's distance from the node, at these times,
within the limits of 6000 years, either before or after any given
year in the 18th: century ; and the mean time of any. New or
Full Moon in any month of the year after March, by means of .
the third and fourth tables, within the same limits, as shown in
the precepts for calculation.
Thus it would be a very easy matter to calculate the time of
any New or Full Moon, if the Sun and Moon moved equably ia
all parts of their orbits. But we have already observed that
their places are never the same as they would be by equable
motions, except when their mean anomalies are cither 0, or 6
signs ; and that their mean places are always forwarder than
than their true places, while their anomalies are less than 6
signs ; and their true places are forwarder than the mean, while
the anomaly is more.
Hence it is evident, that while the Sun's anonaly is less than
6 signs, the Moon will overtake him, or be opposite to him,
sooner than she could if his motion were equable ; and later
while his anomaly is mora than 6 signs. The greatest differ-
ence that can possibly happen between the mean arid true time
of New or Full Moon, on account of the inequality of the Sun's
motion, is 3 hours 48 minuets 28 seconds ; and this is when
the Sun's anomaly is either 3 signs 1 degree, or 8 signs 29 de*
grees ; sooner in the first cas-^, and later in the last* lit. all
I
146 LOGARITHMICK
other signs and degrees of anomaly, the difference is gradually
lets, and vanishes when the anomaly is either 0, or 6 signs.
The Sun is in his apogee on the 30th. of June, and in his
perigee on the 30th. of December, in the present age ; so that
he is nearer the earth in our winter than in our summer. The
proportional difference of the Sun's apparent diameter at these
limes, is as 983 to 1017.
The Moon's orbit is dilated in winter, and contracted in sum-
mer. The greatest difference is found to be 22 minuets 39
seconds ; the lunations increasing gradually in length while
the Sun is moving from his apogee, and decreasing in length
while he is moving from his perigee to his apogee. On this
account the Moon will be later in coming to her conjunction
with the Sun, or being in oppositions him,* from December till
June, and sooner from June to December, than if her orbit had
continued of the same size all the year round.
As both these differences depend on the Sun's anomaly, they
may be fitly put together into one table and called The annual,
orjirat equation for reducing the mean to the true syzygy.\ (See
Table VII.)
This equational difference is to be subtracted from the time
of mean New or Full Moon when the Sun's anomaly is less
than 6 signs, and added when the anomaly is more.
At the greatest, it is 4 hours 10 minuets 57 seconds, Viz. 3
hours 48 minuets 28 seconds, on account of the Sun's unequal
motion, and 22 minuets 29 seconus, on account of the dilation
of the Moon's orbit.
This compound equation would be snmcient for reducing the
moan time of New or Full Moon to the true time, if the Moon's
orbit were of a circular form, and her motion exactly equable in
it. But the Moon's orbit is more elliptical than the Sun's, and
her motion in it is so much the more unequal. The difference
is so great, that she is sometimes in conjunction with the Sun,
or in opposition to him, sooner by 9 hours 47 minuets 54 sec-
onds, than she would be if her motion were equable ; and at
other times as much later. The former happens when her
mean anomaly is 9 signs 4 degrees, and the latter when it is 2
signs 26 degrees. See Table IX.
At different distances of the Sun from the Moon's apogee, the
figure of the Moon's orbit becomes different.
It is longest of all, or most excentrick, when the Sun is in
* The term conjunction, when it respects the relation of the.
Moon to the Sun, signifies New Moon, or Change ; and oppo-
sition is used to signify the place of the Moon at her full.
tThe word syzygy siguifies both the conjunction and opposi-
tion of the Sun and Moon.
ARITHMETIC!*. iff
the same sign and degree either with the Moon's apogee or
perigee ; shortest of all, or least exccntrick, when the Sun's
distance from the Moon's apogee is either 3 signs or 9 signs ;
and at a mean state when the distance is either 1 sign 1 5 degrees,
4 signs 15 degrees, 7 signs 15 degrees, or 10 signs 15 degrees.
When the Moon's orbit is at its greatest excentricity, her apo-
geal distance from the Earth's centre is to her perigeal dis-
tance from it, as 1067 is to 933 > when least excentiick, as 1043
is to 457, and when at the mean state, as 1055 is to 945.
But the Sun's distance from the Moon's apogee is equal to
the quantity of the Moon's mean anomaly at the time of New
Moon, and by the addition of 6 signs, it becomes equal in quan-
tity to. the Moon's mean anomaly at the time of Full Moon.
Therefore, a table may be constructed so as to answer all the
various irregualities depending on the different excenlricities
of the Moon's orbit in the syzygies ; and called The second
equation for reducing the mean to the true syzygy, (See Table
IX.) and the Moon's anomaly, when equated by Table^VIH.
may be made the proper argument for taking out this second
equation of time, which must be added to the former equated
time, when the Moon's anomaly is less than 6 signs, and sub-
tracted when the anomaly is more.
There are several other inequalities in the Moon's motion,
which sometimes bring the true syzygy a litle sooner, and at
other times keep it back aiittle later than it would otherwise
be ; but they are so small, that they may be all omitted except
two ; the former of which (See Table X.) depends on the differ-
ence between the anomalies of the Sun and Moon in the syzygies
«nd, the latter, (see Table XI.) depends on the Sun's distance
from the Moon's nodes at these times. The greatest difference
arising from the former, is 4 minuets 58 seconds ; and from
the latter, 1 minuet 34 seconds.
Besides the tables already mentioned, there are various others
annexed in the following, to facilitate the labour of astronomi-
cal calculations* and will be treated of in their proper place.
+ •*■ ■&
* •$ * i
■%■ •$• -£
•*• ■*•
■*■ * .
.St.
TABLES
ARITHMET1CK.
149
TABLES
FOR CALCULATING THE TRUE TIME OF NEW
AND FULL MQOA'S AND ECLIPSES.
TABLE I. The mean Time of New Mjoh in March, Old Style,
with the mean Anomalies of the Sun and Moon, and the Sun 9 *
Mean Distance from the Moon's Aacinding Node, from A. D
1700 to A. D. 800 inclusive
O
S3*
1700
1701
1702
1703
1704
Mean iNcw
Moon
in March.
D. H. M. S.
8 16 11
27 13.44
16 22 32
6 7 2i
24 4 53
dun s mean
Anomaly.
O f tf
Moon's mcuij Sun's mean
Anomaly
1705
1706
1707
1708
1709
1710
1711
1712
1713
1714
13 13 42
2 22 31
31 20 3
10 4 52
29 2 25
258
59
418
188
il'L
348
118
509
278
79
lisiance from
the Node.
19 58 48 1 22 30 3?
8 20 59 28
27 36 5l'll 7
16 52 43 9 17
5 14 54' 8 23
55 47
43 52
20 57
42 7
8
8
9
14 31 7
23 14 8
1 16 55
9 19 42
18 2 43
24 30 47i 7 3 9 2
13 46 39 5 12 57 7
2 8 50 4 18 34 13
21 24 48 2 28 22 18
9 46 54' 2 3 59 24
18 11 13
7 20 2
25 17 34
15 2 23
4 11 12
1715
1716
1717
1718 19
1719
438
208
599
36 8
13 8
23
11
1
8 44
17 33
2
23
22
54
8 43
1720 27
172116
1722 5
1723 24
1724 13
1725 2 15
1726 21 12
10 21
1727
1728
1629
6
15
23
fil
6
16
4
ly
38
29 2 471
18 18 3910
6 40 51 9
25 56 431 8
15 12 35 6
3 34 47
22 50 39
12 6 32
28 44
13 47 30
23 35 36
29 12 42
9 47
18 48 52
26 5 30
4 8 17
12 51 18
20 54 5
29 37 6
8 19 44 3711
1
1
2
3
3_
24 25 571 4
4 14 2 4
14 2 8 5
19 39 13, 6
29 27 18 6
7 39 54
15 42 41
14 25 43
2 28 30
10 31 17
19 14 18
27 17 5
5 19 52
14 2 54
22 5 41
53 14
25 54
14 31
28 18
18 3
1730 7
1731 26
173V 14
1733 4
1734 23
3 78
35 47 9
24 23 8
57 * 3 9
45 40 8
8"
8
27
16
5
24
6
22
38
16
4911
5 4 241 8
14 52 29 8
24 40 34 8
17 40 9
10 5 45 10
48 43
8 51 29
16 54 16
25 37 18
3 40 5
13 32 291 3 19
1 54 41 2 25
21 10 34 1 5
9 52 46 10
28 48 3910 20
5010
5611
1 11
34 16
6 56
55 33
44
16 4S
18 4 31 9
6 26 42 8 6
25 42 34 6 15
U 58 26 4 25
3 20 38 4 1
17
23
28
S3
39
11 42 52
20 25 54
28 28 41
7 11 42
15 14 29
23 17 16
2 17
10 3 4
18 5 51
26 48 51
17*o|12 10 3 25 8 22 36 30 2 11 10 44
175 C 13 54 2 8 11 52 22 20 58 49
1737'. 19 16 26 42^9 14 3411 26 35 55
1738 9 1 15 18 8 19 30 2610 6 24
1739127 22 47 5819 7 52 38! 9 12 1 6
20
5 4 51 40
5 12 5i 27
6 21 37 29
6 29 40 16
8 8 23 18
150
LOGARITHMIC*
TABLL
1.
continued. Old Slytt.
<
Mean New
b
un b
mean
Moon's mean
Sun's mean
o
Moon
Anomaly.
Anomaly.
distance from
O
1 E
li
D.
i March.
n
the Node.
H. M. S
s
/
s ' f '\ s ' ''
1740
16
7 36 34
3
27
8
30
7 21 49 Hi 8 16 26 5
1741
5
16 25 11
8
16
24
22
6 1 37 16 8 24 28 52
1742
24
13 57 52 9
4
46
34
5 7 14 2210 3 11 54
1743
13
22 46 27,8
24
2
27
3 17 2 27,10 1 14 41
1744
1745
2
7 35 43
13
18
20
1 26 50 32
10 19 17 28
11 28 30
21
5 7 449
1
40
32
1 2 27 38
1746
10
13 56 2t)8
20
56
24
11 12 15 43
6 3 17
1747
29
11 29 09
9
18
3610 17 52 49
1 14 46 19
1748
17
20 17 368
28
34
28
8 27 40 54
1 22 49 5
1749
1750
7
5 6 13-8
17
50
20
7 7 28 59
[ 2 ♦ 51 52
26
2 38 53,9
6
12
32
6 13 6 5
3 9 34 53
1751
15
11 27 29|8
25
28
24
4 23 54 10 3 17 37 40
1752
3
20 16 6 3
14
44
16
3 2 42 15
3 35 40 27
1753
22
17 48 45 9
O
6
28
2 6 19 21
S 4 23 28
1754
1755
12
2 37 228
22
22
20
11 7 26
5 12 26 15
1
11 25 598
11
38
12
10 27 55 31
5 20 29 2
1756
19
8 58 389
24
10 3 32 37
6 29 12 3
1757
8
17 47 15 8
19
16
16
8 13 20 42
7 7 14 50
1758
27
15 19 549
7
38
88
7 28 57 4fc
8 15 57 52
1759
1760
17
8 318
26
54
20
5 28 45 54
8 24 39
5
8 57 8 8
16
10
12
4 « 34 6
9 2 3 26
1761
24
6 29 479
4
32
24
3 14 11 6
10 10 46 27
1762
13
15 18 248
23
48
16
1 23 59 11
10 18 49 14
1763
3
7 1
8
13
4
8
3 47 16
10 26 52 1
1764
1765
20
21 39 40
9
1
26
42
20
13
11 9 24 21
5 35 2
10
6 28 17
8
20
9 19 12 26
13 37 49
1766
29
4 56
9
9
4
20
8 24 49 35^
1 22 20 51
1767
18
12 49 33
«
28
20
17
7 4 37 37
2 23 38
1768
6
21 38 10
8
17
36
9
5 14 25 42
2 8 26 25
1769
1770
25
19 . 10 40
9
5
58
21
4 20 2 48
1 3 17 9 27
15
3 59 26
8
25
14
13
2 29 50 53
3 25 12 14
1771
4
12 48 2
8
14
30
5
1 9 38 58
4 3 15 1
1772
22
10 20 43
9
2
52
17
15 16 4
5 11 58 3
1773
11
19 9 19
8
22
8
9
10 25 4 9
5 20 50
1774
1
8 57 55
8
.11
24
1
9 4 52 14
5 28 3 37
1775
20
1 30 25
8
29
46
13
8 10 29 20
7 6 49 38
1776
*
10 19 12
8
19
2
5
6 20 17 25
7 14 49 25
1777
27
7 51 51
9
7
24
17
5 25 54 31
8 23 32 26
1778
16
16 40 28
8
26
40
9
4 5 42 86
9 1 35 13
177Q
6
I 29 4
8
15
56
1
2 15 30 41
9 9 38
1780
23
23 1 44
9
4
18
13
1 21 7 47
10 18 21 1
1781
13
7 50 21
8
23
34
5
55 52
10 26 23 48
1782
2
16 38 57*
8.
12
49
58
10 10 43 57
11 4 26 35
1783
21
14 11 37
9
1
12
10
9 16 21 3
13 9 36
1784
1785
9
28
23 13
3
20
28
3
7 26 9 8
21 12 23
U0 32 56[9
9
50
15
7 1 46 14
1 29 55 25
"' 17861
18
5 21 30'8
28
6
71 5 11 34 19
2 7 58 12
ARITHMET1CK.
151
1
TABLE
1. concluded.
Old Styl
c.
*:
M
tun
New
Sun's, mean
Moon'
s meai.
Sun's .mean
o
Moon
Anomaly
' Anomaly.
'distance from
O
in
March.
s
O
the >
s
J ode.
D.
H.
M. S
s
'.
//
/
//
/ a
1787
7
14
10 6 8
17 21
59
3
21
22
24
2
16
59
1788
25
11
42 46 9
5 44
11
2
26
59
30
3
24
44 1
1789
14
20
31 238
25
3
1
6
47
35
4
2
46 48
1790
4
5
19 59 8
14 15
5511
16
35
40
4
10
49 35
1791
23
2
52 39
9
8
2^ 38
21 53
7
59
10
22
12
46
5
19
32 37
1792
11
11
41 15.
9
2
U
52
5
27
35 24
1793
30
9
13 559
10 16
11 8
7
37
58
7
6
18 26
1794
19
18
2 32'8
29<> 32
3 6
17
26
4
7
14
21 13
1795
9
2
51 88
18 47
5S, 4
27
14
9
7
22
24
1796 27
17971 16
1798) -5
9
23 48
12 24.
9
7 10
71 4
2
51
14
9
1
7 1
8
26 tS
59 2
12
3*
19
9
9
9 48
18
1 18
15 41
51'
22
27
25
9
17
12 35
1799,24
15
23 41 9
4 4
311
28
4
31
10
25
55 37
1800]
13
22 17
8
23 19
5510
7
52
36
11
3
58 22
TAULk 11. Mean New AJoon, <Jfc.iu iviaixh, *viu; o;
yw, ji-om
A. D. 1752 to A.
D. 19 r 0. *
^
Meat
New
Sun's
mean
Moon's meai.
■'JJUII'ji
mean
P
Moon
Anomaly.
.
Anomaly.
distant
:e from
O
in M
arch.
the Node.
-i
CD
P.
H.
M. S
s
/ a
s
'
ii
s
O
i a
1752
14
20
16 6
8
14
44 16
3
2 42
15
3
25
4U 27
1753
4
5
4 42
8
4
8
1
12 30
20
4
3
43 14
1754
23
2
37 22
8
22
22 20
18 7
26
5
12
26 15
1755
12
11
25 59
8
11
38 12
10
27 55
31
5
20
29 2
1756
30
8
58 38
9
24
10
3 32
37
6
29
12 3
1757
19
17
47 IS
8
19
16 16
8
13 20
42
7
7
14 50
1758
9
2
35 51
8
8
32 8
9
23 8
47
7
15
17 38
1759
28
8 31
8
26
54 20
5
28 45
54
8
24
39
1760
16
8
57 8
8
16
10 12
4
8 34
9
2
3 26
1761
1762
5
17
45 44
8
5
26> 4
2
18 22
5
9
10
6 13
24
15
18 24
8
23
48 16
1
23 59
lljlO
16'10
18
49 14
1763
14
7 1
8
13
4 8
3 47
26
52 1
1764
2
8
55 36
8
2
20.
10
13 35
21
11
4
54 48 t
1765
21
6
28 17
8
20
42 13
9
19 12
26
13
37 49
1766
10
15
16 53
8
9
58 5\ 7
29
31
21
40 37
1767
29
12
49 33
8
28
20 17
7
4 37
3/
2
~3 38
1768
17
21
38 9
8
17
36 9
5
14 25
42
2
8
26 25
1769
7
6
26 46
8
6
52 1
3
24 13
47
2
16
29 13
1770 26
3
59 26
8
25
14 13 2
29 50
53
3
25
12 14
1771 Il5
12
48 2
8
14
30 5' 1
9 38
58 4
3
15 1
17721 3
117SI22
21
36 39
8
3
45 57U
19 27
3 4
11
17 48
19
9 19
8
ae
8 9 10
25 4
9 5
20
50
177412
3
57 55
8
n
24 1 9
4 52
14 5
28
3 37
17751 1
12
46 31
8
39 53 7
14 40
19 6
6
fc *i\ \
1776
19
10
19 12
8
19
2 6 6
20 17
2,5 7
\4i
AS) ^5
452
LOGARITHMIC*
*
fAHLE 11. continued.
JVew Style.
^
Mean iNcvv
buu'fc
mean
Moon's meant Sun's mean
c
Moon.
Anomaly.
Anomaly, (distance from
O
Or
in March.
the Node.
D H.
M. S.
s
f rf
s
' h
s O
/ tr
1777
8 19
7 48
8 8
17 5f\ 5
5 o0
7 22
52 12
1778
27 16
40 28
8 26
40 9 4
5
42 36
9 1
35 13
1379
17 1
29 4
8 15
56 1 2
15
30 41
9 9
38
1780
5 10
15 3
8 5
5 54
25
22 58
9 17
36 12
1781
24 7
47 40
8 23
28 4]
1
9
10 26
19 14
1782
13 16
36 14
8 12
43 55*10
10
48 18
11 4
22 1
1783
3 1
24 48
8 1
59 47
8
20
36 28
11 12
24 49
1784
20 22
37 25
8 20
21 57
7
26
13 39
21
7 50
1785
10 7
45 59
8 9
37 48
6
6
1 49
29
10 38
1786
29 5
18 36
8 27
59 58
5
11
38 59
2 7
53 39
1787
18 14
7 10
8 17
15 50
3
21
27 9
2 15
56 26
1788
6 22
55 45
8 6
31 40
2
1
15 19
2 23
59 14
1789
25 20
28 22
8 24
53 51
1
6
52 30
4 2
42 15
1790
15 5
16 56
8 14
9 42
11
16
40 39
4 10
45 3
1791
4 14
5 30
8 3
25 33
9
26
28 49
4 18
47 50
1792
22 11
38 7
8 21
47 44
9
2
6
5 27
30 52
1793
11 20
26 41
8 11
3 35
7
11
54 10
6 5
33 39
1294
30 17
5*. 18
8 29
25 45
6
17
31 20
7 14
16 41
1795
20. 2
47 53
8 18
41 36
4
27
19 30
7 22
19 28
1796
8 11
36 27
8 7
57 28
3
7
7 40
8
9 9
22 16
5 17
1797
27 9
9 4] 8 26
19 38
2
12
44 51
1798
16 17
57 38 8 15
35 29
22
33
9 17
8 5
1799
6 2
46 12 8 4
51 20
11
2
21 10
6 25
10 52
1800
25
18 49 8 23
13 30
10
7
58 21
11 3
53 54
1801
14 9
7 23! 8 12
29 22
8
17
46 31
11 11
56 41
1802
3 17
55 5b
8 1
45 13
6
27
34 41
11 19
59 29
1803
22 15
28 35
8 20
7 25
6
3
11 51
28
42 30
1804
11
17 fi
8 9
23 14
4
13
1
1 6
45 18
1405
9
5 43
8 28
39 5
2
22
48 11
1 14
48 $
1806
1807
19 6
38 20
8 17
1 16
1
28
25 21
2 23
3 1
31 7
33 54
b 15
26 55-, 8 6
17 7
8
13 31
1808
26 12
59 311 8 24
39 17
11
13
50 42
4 10
16 56
1809
15 21
48 5\ 8 13
55 8
9
23
38 52
4 18
19 43
1810
5 6
36 40' 8 3
10 59
8
3
27 2
4 26
22 31
1811
24 4
9 17| 8 21
33 10
7
9
4 12
6 5
5 32
1812
12 12
57 51 8 10
49 1
5
18
52 22
6 13
8 20
1813
1 21
46 25 8
4 52
3
28
40 32
6 21
11 7
1814
20 19
19 2; 8 18
27 3
3
4
17 43
7 29
54 9
1815
10 4
7 37 8 7
42 54
1
14
5 52
8 7
56 56
1810
2» 1
40 14| 8 26
5 4
19
43 3
9 16
39 58
1317 17 10
28 48 8 15
20 55
10
29
31 13
9 24
42 45
1818
6 19
17 22 8 4
36 46
9
9
19 23
10 2
45 33
1819
25 16
51 14| 8 23
5 44
8
14
41 17
11 11
29 55
182J
,14 1
38 33 1 8 12
14 48
6
24
44 43
11 19
31 22
18211 3 10
27 7 8 1
30 39
5
4
32 53jll 27
34 9
1822|22 7
49 45! 8 19
52 50
4
10
10 34| 1 6
17 10
1823
111 16
48 19
I 8 9
8 41
2
19
58 13
1 14
19 58
ARITHMETIC*.
153
TABLE 11. continued. JYcv> Style.
o
o
sr
CO
Mean New
Moon
in March.
Sun's Mean
Anomaly.
Moon's meai,
Anomaly.
Sun's mean
distance from
the Node.
D. H. M. S
s
' "
so'"
s O ' "
1824
1825
1826
1827
1828
29 14 20 56
18 23 9 SO
8 7 59 4
27 5 30 41
15 14 19 15
8
8
8
8
8
27 30 51
16 46 42
6 2 33
24 24 44
13 40 35
1 25 35 24
5 33 34
10 45 11 44
9 20 48 54
8 37 4
2 23 2 59
3 1 5 47
3, 9 8 34
4 17 51 36
4 25 54 23
18291
1830:
1831
1832
1833
1834
1835
1836
1837
1838
4 23 7 50
23 20 40 27
13 5 29 1
1 14 17 35
20 11 50 12
8
8
8
8
8
2 56 26
21 18 36
10 34 27
29 50 19
18 12 29
6 10 25 15
5 16 2 55
3 25 50 34
2 5 38 44
1 11 15 55
5 3 7 11
6 12 40 12
6 20 3
6 28 45 47
8 7 28 49'
9 20 38 46
28 18 11 23
17 2 59 58
6 11 48 32
25 9 21 9
8
8
8
8
8
7 28 20
25 50 31
15 6 22
4 22 13
22 44 23
11 21 4 4
10 26 41 15
9 6 29 25
7 16 17 35
6 21 54 46
8 15 31 36
9 24 14 38
10 2 17 25
t0 10 20 13
U 19 3 14
1839
1840
1841
1842
1843
14 18 9 43
3 2 58 17
22 30 54
11 9 19 28
30 6 52 6
8
8
8
8
8
12 14
1 16 5
19 38 16
8 54 7
27 16 17
5 1 42 55
3 11 31 5
2 17 8 16
26 56 26
2 33 36
11 27 6 2
5 8 49
1 13 51 51
1 21 54 38
3 37 40
1844
1845
1846
1847
1848
18 15 40 40
8 29 14
26 22 1 51
16 6 50 25
4 15 38 59
8
8
8
|8
16 32 8
5 47 60
24 10 10
13 26 1
• 2 41 5
10 12 21 46
8 22 9 56
7 27 47 7
6 7 35 16
4 '17 23 26
3 8 40 27
3 16 43 15
4 25 26 16
5 3 29 4
5 11 31 51
1849
1850
1851
1852
1853
23 13 11 36
12 22 11
2 6 49 48
20 4 21 27
9 13 10 4
8
8
7
8
8
21 4 3
10 19 54
29 35 46
17 57 58
7 13 5C
3 23 -37
2 2 48 47
1 12 36 52
'11 18 13 58
I 9 28 2 3
6 20 14 53
6 28 17 40
7 6 20 27
8 15 3 28
8 23 6 15
1854
1855
1856
1857
1858
28 10 42 43
17 19 31 20
► 5 4 19 57
'24 1 52 36
113 10 41 13
i 8
8
8
8
8
25 36 2
14 51 54
A 7 At
22 29 5i
11 43 5C
9 3 39 9
t 7 13 27 14
> 5 23 15 19
\ 4 28 52 25
V 3 8 40 30
10 1 49 16
10 9 52 3
10 17 54 50
11 26 37 51 •
4 40 38
18591 2 19 29 49
186020 17 2 29
1861J10 1 51 6
1862:28 22 23 45
8
8
8
8
1 1 42| 1 18 28 35
19 23 54' 24 5 41
8 39 4611 3 53 4(
27 1 5810 9 30 55
12 43 25.
1 21 46 26
1 29 49 13
3 8 32 14
1863.17 7 12 22
1864| 5 16 59
1865 24 13 33 38
186613 22 22 15
8
8
1 8
8
16 17 50 8 19 18 57
6 33 24' 6 29 7 S
24 55 54' 6 4 14 I
14 11 46 4 14 2 i:
3 16 35 1
3 24 37 48
5 5 3 20 49
* 5 11. 23 36
1867j 3 7 10 51
186821 4 43 31
1869.10 13 31 8
1870*28 11 4 47
8
8
t 8
' 8
3 27 38. 2 23 50 1*
21 49 50 1 29 27' 24
11 5 4210 9 15 2*
29 27 5411 14 52 3.
J 5 19 26 23
I 6 28 9 24
) 7 6 12 11
5 8 14 55 12
154
LOGARITHMIC*
TAULk
. 11
. concluded, JS'ew
Style.
<
Mean New
Sun's
Mean
Moon 9
a meaii
Sun's mean
Q
Moon
Anomaly.
Anomaly.
distance from
o
**•
in March.
tl
s
ie Node.
D. H.
M.
S.
s
r
tf
s O
/
fi
' "
18/1
17 12
53
24
8
18
43
46
9 24
40
40
8
22 57 59
1872
5 4
42
8
7
59
38
8 4
28
45
9
1 46
1873
24 2
15
40
8
26
21
50
7 10
5
5110
9 43 47
1874
13 11
4
17
8
15
37
42
5 19
53
56
10
17 46 34
1875
lb/o
2 19
52
53
8
4
53
34
3 29
42
1
10
25 49 21
20 11
25
33
8
23
15
46
3 5
19
7
4' 32 22
1877
10 2
14
10
8
12
31
38
1 15
7
12
12 35 9
1878
28 23
46
49
9
53
50
20
44
18
21 18 10
1879
18 8
35
26
8
20
9
4211
32
23
1
29 20 57
1880
1881
6 17
24
2
8
9
25
34| 9 10
20
28
2
7 23 44
25 14
56
42
8
27
47
46j 8 15
57
34
3
16 6 45
1882
14 23
45
19
8
17
3
38 6 25
45
39
3
24 9 32
1883
4 8
33
55
8
6
19
30 5 5
33
44
4
2 12 19
1884
22 6
6
35
8
24
41
42
4 11
10
50
5
10 55 20
1885
11 14
55
11
8
13
57
34
2 20
58
55
5
18 58 7
188b
23
43
48
8
3
13
26
1
47
5
27 54
1887
18 21
16
28
8
21
35
38
6
24
6
7
5 43 55
1888
7 6
5
4
8
10
51
3010 16
12
11
7
13 46 42
1889
26 3
37
44
8
29
13
42 9 21
49
17
8
22 29 43
1890
1891
15 12
26
21
8
18
29
34J 8 1
37
22
9
32 30
4 21
14
57
8
7
45
26' 6 11
25
27
9
8 35 IX
1892
22 18
47
37
8
26
7
38 4 17
2
33
10
17 18 ia
1893 12 3
36
13
8
15
23
30 2 ' 26
50
38
10
25 21 5
1894
1 15
24
50 8
4
39
22 1 6
38
43
11
3 23 52
1895
20 9
57
30
8
23
1
34! 12
15
49
12 6 53
1896
8 18
46
6
8
12
17
2610 22
3
54
O
29 9 40
1897
27 15
18
47
9
39
38 9 21
41
1
28 52 41
1898
17
7
23
8
19
35
30 8 7
29
5
2
6 55 28
' 1899
6 8
56
8
9
11
22 6 17
17
10
2
14 58 15
1900
24 6
28
40 8
27
33
34| 4 22
54
16|
3
23 41 16
AKITHMETICK.
)?5
TABLE III. Mean Anomalies^ and Sun 9 * mean Distance from
the Nodcyfor 13-J mean Lunations.
No.
1
2
3
4
5
Mean
Lunations.
Suns mean
Anomaly.
Moon's meari
Anomaly
bun's mean
distance from
the Node.
D H. M. S
s ' "
s '
//
s ' "
29 12 44 3
59 I 28 6
88 14 12 9
118 2 56 12
147 15 40 15
29 6 19
1 28 12 39
2 27 18 58
3 26 25 17
4 25 31 37
25 49
1 21 38
2 17 27
3 13 16
4 9 5
1
1
2
2
1 40 14
2 1 20 28
3 2 42
4 2 40 56
5 3 21 10
6
7
8
9
10
1*7 4 14 It*
206 17 8 21
236 5 52 24
265 18 56 27
295 7 20 30
5 24 37 56
6 23 44 15
7 22 50 35
8 21 56 54
9 21 3 14
5 4 54
6 43
6 26 32
7 22 21
8 18 10
3
3
3
4
4
6 4 1 24
7 4 41 38
8 5 21 52
9 6 2 6
10 6 42 20
11
12
13
324 20 4 33
354 8 48 30
38*3 21 32 40
10 20 9 33
11 19 15 52
18 22 12
9 13 59
10 9 48
11 5 37
5
5
6
11 7 22 34
8 2 47
1 8 43 1
.14 18 22 2
14 33 iO
6 12 54
3ul
15 20 7
156
LOGARITHMICK
1ABLE IV. The Day 9 of the Year, reckoned from the begin-
ning of March.
O
to
s
p
O
ar
2
p
>
e
<*?
c
r
C/3
B
73
O
r*
O
cr
71
3
<
B
0-
7*
246
247
248
249
250
a
rt
B
276
277
278
279
280
*-•
P>
S
C
P
~$
307
308
309
310
311
cr
e
338
359
340
341
342
1
2
3
4
5
1
2
3
4
5
32
<■» r»
OO
34
35
36
62
63
64
65
66
93
94
95
96
97
123
124
125
126
127
154
155
156
157
158
185
186
187
188
189
215
216
217
218
219
6
7
a
9
10
6
7
8
9
10
37
38
39
40
41
67
68
69
70
71
98
99
100
101
1C2
128
129
130
151
132
159
160
161
162
163
190
191
192
193
194
220
221
222
223
224
251
252
255
254
255
281
282
283
284
285
312
315
314
315
316
343
344
345
346
347
448
349
350
351
352
11
12
13
14
15
11
12
13
14
15
. 42
43
44
45
46
72
73
74
75
76
103
104
105
106
107
133
134
135
136
137
164
165
166
167
168
195
196
197
198
199
225
226
227
228
229
256
257
258
259
260
286
287
288
289
290
317
318
319
320
321
16
17
18
19
20
16
17
18
19
20
47
48
49
50
51
77
78
79
80
81
108
109
110
HI
112
138
139
140
141
142
169
170
171
172
173
200
201
202
203
204
230
231
232
233
234
261
262
263
264
265
291
292
293
294
295
296
297
298
299
300
301
302
503
304
305
306
322
323
324
325
326
327
328
329
330
331
332
383
334
335
336
337
353
354
355
356
357
358
359
360
361
362
363
364
365
366
21
22
23
24
25
26
27
28
29
30
31
21
22
23
24
25
52
53
54
55
56
82
83
84
85
86
113
114
115
lie
117
143
144
145
146
147
174
175
176
177
178
205
206
207
208
209
235
236
237
238
239
266
267
268
269
270
26
27
28
29
30
31
57
58
59
60
61
87
88
89
90
91
92
lib
W9
120
121
122
148
149
150
151
152
153
179
180
18i
182
185
184
210
211
212
213
214
240
241
242
243
244
245
271
272
273
274
275
ARITHMETICS.
1ST
TABLE V. Mean Lunations from 1
to 100000.
Lunat.
Days Deci. parts.
Days Hou.
M. S. Th.
Fo.
1
29-530590851080
= 29
12
44 3 2
58
2
59-061181702160
59
1
28 6 5
57
3
88-591772553240
88
14
12 9 8
55
4
118-122363404320
118
2
56 12 11
55
5
147-652954255401
147
15
40 15 14
52-
6
177-183545106481
177
4
24 18 17
50
7
206-714135957561
206
17
8 21 20
48
8
236-244726808641
236
5
52 24 23
47
9
265-775317659722
265
18
36 27 26
45
10
295-30590851080
295
7
20 30 29
43
20
590-61181702160
590
14
41 59
26
30
885-91772553240
885
22
1 31 29
10
40
1181-22363404320
1181
5
22 1 58
53
50
1476-52954255401
1476
12
42 32 28
36
60
1771-83545106481
1771
20
3 2 58
19
70
2067-14135957561
2067
3
23 33 28
2
80
2362-44726808641
5362
10
44 3 57
46
90
2657 75317659722
2657
18
4 34 27
29
100
29530590851080
2953
1
25 4 57
12
200
5906.1181702160
5906
2
50 9 54
24
300
, 88591772553240
8859
4
15 14 51
36
400
11812-2363404320
11812
5
40 19 48
48
500
14765-2954255401
14765
7
5 24 46
600
17718-3545106481
17718
8
30 29 43
12
700
20671-4135957561
20671
9
55 34 40
24
800
23624-4726808^641
23624
11
20 39 37
36
x 900
1000
26577-5317659722
26577
12
45 44 54
46
29530-590851080
29530
14
10 49 32
2000
59061-181702160
59061
4
21 39 4
3000
88591-772553140
88591
18
32 28 36
4000
1 18122*36*404320
118122
3
45 18 8
5000
147652-954255401
147652
22
54 7 40
O
6000
177183-545106481
177183
13
4 57 12
7000
206714-135957561
206714
3
15 46 44
8000
236244-726801641
236244
17
26 36 16
9000
265775-317659722
265775
7
37 25 48
10000
295305-90851080
295305
21
48 15 20
20000
59061 181702160
590611
19
56 30 40
30000
885917 72553240
885917
17
24 46
40000
1 188223-6S404320
1188223
15
13 1 20
50000
1476529-542554U1
1476529
13
1 16 40
60000
1771835-45106481
1771835
10
49 32
70000
2067141-35957561
2067141
8
37 47 20
80000
2362447-26808641
2362447
6
25 2 40
90000
2657753-17659722
2657753
4
14 18
6
100000
2953959-0851080
2953959
2
2 33 20
o
21
i
J58
L06ARITHMICK
~ _ . »-.- , i , i Ml. 1 1
TABLE Y r I. The Jirst mean Aew Moott> with the mean Ano-
malies of the Sun arrd Moon, and the Sun's mean Distance, ,
from the Ascending Nodc^ nezt after complete Centuries, of '
Julian Years.
Luna-
tions.
First 1!
Vcw Moon.
Sun's meuti|
Anomaly.
s o ''
Ylon'biiicaii
Anomaly.
Sun from
Node.
D.H. M.S.-
so '
s
O '
. 1237 1
2474
37U
4948
6185
7422
8658
9895
100
200
300
400
4' 8 10 52
8 16 21 44
13 32 37
17 8 43 29
3 2i
« 42
t) 10 3
13. 24
8 15 22
5 44
1 16 6
10 1 28
4
9
1
6
19 27
8 55
28 2%
17 49
500
600
700
800
21 16 54 21
26 1 5 14
20 32 3
5 4 42 $5
16 46
20 7
11 24 22
11 27 34
6 16 50
3 2 12
10 21 45
7 7 7
11
3
7
7 16
26 44
15 31
4 58
11132
12369
15606
14843^
16080
17316
18553
19790
21027
22264
23501
24738
25974
27211
28448
29685
3U922
32159
33396
34632
yoo
1000
1100
1200
9 12 53 47
13 21 4 40
18 5 15 32
22 13 26 24
14
4 25
7 46
11 7
3 22 29
7 51
8 23 13
5 8 35
4
9
2
6
V4 25
13 53
3 20
22. 47
12 15
X 2
20 29
9 56
1300
1400
1500
1606
26 21 37 16
1 17 4 6
6 1 14 58
10 9 25 50
14 28
11 18 43
11 22 4
11 25 25
1 23 57
9 13 30
5 28 52
2 14 14
11
3
7
1700
1800
1900
2000
14 17 36 42
19 1 47 35
21 9 58 27
27 18 9 19
11 2<i 46
2 8
5 29
8 50
10 29 36
7 14 58
4 20
15 42
4
9
2
6
29 23
18 51
8 18
27 45
2100
2200
2300
2400
2 13 36 ^
6 21 47 1
11 5 57 53
15 14 8 45.
11 13 5
11 16 26
11 19 47
11 23 8
8 5 15
4 20 57
1 5 59
9 21 21
10
3
7
16 32
6
25 27
14 54
2500
2600
2700
2800
19 22 19 3H
24 6 30 30
28 14 41 22
£ 10 8 11
11 26 29
11 29 50
3 11
11 7 76
6 *6 43
2 22 4
11 7 26
6 26 59
■ 5
9
2
6
4 22
23 4©
13 16
2 3
35869
37106
38343
39580
2900
3000
3100
3200
7 18 19 3
12 2 29 56
16 10 40 48
J 20 18 51 40
11 10 47
11 14 8
11 17 30
11 20 51
3 12 21
11 27 43
- 8 13 5
4 28 27
10
3
8
I
21 30
10 58
25
19 52 J
ARITHMETICS
15?
TABLE VI. concluded.
■ ■ !»
V<J t-|
Fust
Sun
*s mean
Mon'smeanl Sun from
Luna-
tion a.
S&.'
New Moon-
Anomaly. <
. Anomaly. J Node.
* g
D. H. M. S
s
o
t
s
o
/
s
O '
4081 7
3300
25 3 2 33
11
24
12
1
13
49
5
9 20
42054
3400
29 11 13 25
11
27
33
9
29
11
9
28 47
43290
S50Q
4 6 40 14
11
1
48
5
18
44
1
17 34
44527
3600
8 14 51 6
11
5
9
2
4
6
6
10
7 1
26 29
45764 |
3700
12 23 1 59
U
8
30
10
19
28
47001
3800
17 7 12 51
U
11
51
7
4
50
3
15 56
48238
3900
21 15 23 43
11
15
12
3
20
12
8
5 23
49475
4000
25 23 34 35
11
18
33
5
34
24 50
50/11
4100
19 1 27
10
22
4o
7
25
7
4
13 %r
51948
4200
5 3 12 17
10
26
9
4
10
29
9
3 5
53185
4300
6 11 23 *9
-10
29
31
25
51
1
22 32
54422 4400
13 19 34 1
11
2
52
9
11
33
6
11 59
55659 1 4500
18 3 44 54
11
6
13
5
26
35
11
1 27
56896 4600
22 11 55 46
11
9
34
2
11
57
3
20 54
58133 4700
26 20 6 38
11
n
55
10
27
19
8
10 ?1
59369 1 4800
1 15 33 27
10
10
17
9
6
16
52
U
29 8
6U606
4900
5 23 44 20
20
21
o
2
14
4
18 36
61843
5000
10 7 55 12
10
23
52
li
17
30
9
8 3
63080
5100
14 16 6 4
10
27
13
8
2
58
1
27 30
64317
5200
19 16 56
11
34
4
18
20
6
19 57
65354
5300
23 8 27 49
11
3
J>5
1
o
42
11
6 25
66791
5400
27 16 38 41
id
7
16
9
19
4
2
25 52
68028
5500
2 12 5 30
10
11
31
5
8
S7
7
.14 39
69265
5600
6 20 16 22
10
14
52
1 1
23
59
4 6
23 34
70502 1 5700
11 4 27 15
10
18
14
10
9
21
71739 5803"
15 12 38 7 10
21
35
6
24
43
9
13 1
72976 5900
19 20 48 59 10
24
56
3
10
5
2
2 28
74212 6000 24 4 59 52 1 10
28
17 U
25
27
6
21 56
' h Ov. found mean Lunation (which we
* have kept
uy in
:nakin£ these tables) be added 74212 times
to itself, the
sum will amount to 6ooo Julian years, 24
days 4 hours 59
mnutes 51 seconds 4o thirds
; as
reeing with the first part
>Fthe iast line of this table, with
n halt' a
second.
■ "r't
t6*
LOGARITHMICK
TABLE Vll. The annual) or Jirttt Equation of the mean to the
true Syzygy.
Argument
. Sun's mean Anomaly.
Subtract.
a
(ft
CD
Signs.
1
Sign.
1 2
Signs.
H. M.S
3
Signs.
4
Signs.
5
Signs.
S?
n
CO
15
29
28
27
26
25
T4
23
22
21
20
H.
M.S
H.M S
H.M.S
H. MS
H. M. S.
2 3 12
3
35
4 10 53
3 39 30
2 7 45
1
2
3
4
5
4 18
B35
12 51
17 8
21 24
2 6 do
2 10 36
2 14 14
2 17 52
2 21 27
3
3
3
3
3
37 10
39 18
41 23
43 26
45 25
4 10 57
1 4 10 54
4 10 49
4 10 39
4 10 24
3.37 19
3 35 6
3 32 50
3 30 30
3 28 5
i 3 55
2 1
1 56 5
1 52 6
1 48 4
6
8
9
10
11
12
13
14
15
16
17
18
19
20
25 39
28 55
34 11
36 26
42 39
2 25 9
2 28 29
2 31 57
2 35 22
2 38 44
3
3
3
3
3
47 19
49 7
50 50
52 29
54 4
4 10 4
4 9 39
4 9 10
4 8 37
4 7 59
3 25 35
3 23
3 20 20
3 17 35
3 14 49
1 41 1
1 39 56
1 35 49
1 31 41
1 27 31
1
46 52
51 4
55 17
59 27
3 36
2 42 3
2 45 18
2 48 30
2 51 40
2 54 48
3
3
3
3
3
55 35
57 2
58 27
59 49
1 7
4 7 16
4 6 29
4 5 37
4 4 41
4 3 40
3 11 59
3 9 6
3 6 10
3 3 10
3 7
1 23 19
1 19 5
1 14 49
1 10 33
1 6 15
1 1 56
57 36
53 15
48 52
44 28
19
18
17
15
16
14
13
12
11
10
9
8
7
6
5
4
3
2
1
O
«
1
1
1
1
1
7 45
11 53
16
20 6
24 10
2 57 53
3 54
3 3 51
3 6 45
3 9 36
4
4
4
4
4
2 18
3 23
4 22
5 18
6 10
4 2 35
4 1 26
4 12
3 58 52
3 57 27
2 57
2 53 49
2 50 36
2 47 18
2 43 57
21
22
23
* 24
25
1
1
1
1
1
28 12
32 12
36 10
40 6
44 1
3 12 24
3 15 9
3 17 51
3 20 30
3 23 5
T
4
4
4
4
6 58
7 41
8 21
8 57
9 29
3 55 59
3 54 26
3 52 49
3 51 9
3 49 26
2 40 33
2 37 6j
2 33 35
2 30 2
2 26 26
40 2
35 36
31 10
26 44
22 17
26
27
28
29
\ 30
°i
a
00
1
1
1
1
2
47 54
51 46
65 37
59 26
3 12
3 25 36
3 28 3
3 30 26
3 32 45
3 35
4
4
4
4
4
9 o5
10 16
10 33
10 45
10 53
3 47 38
3 45 44
3 43 45
3 41 40
3 39 30
2 22 47
2 19 5
2 15 20
2 11 35
2 7 45
17 50
13 23
8 56
4 29
11
Signs.
10
Signs.
9
Signs.
8
Signs.
7
Signs.
6
Signs.
Add |,
ARITHMETICS
161
iAtiJLk Vlli.
Aquation of the
Muon y 8 mean Anomaly.
Argument. Sun's mean Anomaly.
Subtract.
O
1
2
3
4
i>
O
OB
Signs.
Signs.
Signs.
Sign
s.
n
Signs.
Signs.
o»
o ' ''
'
n
' "
'
'
//
' ''
46
45
1 21 32
1 35
ll 1 23
4
48 19
30
1
1 37
48
10
1 22 21
1 35
2 1 22
14
46 51
29
2
3 13
49
34
1 23 10
1 35
1
1 21
24
45 23
28
3
4 52
50
53
1 23 57
1 35
1 20
32
43 54
27
4
6 28
52
19
1 24 41
1 34
57
1 19
38
42 24
26
5
6
08 6
53
40 1 25 24
1 34
50
1 18
42
40 53
25
9 42
55
1 26 6
1 34
43
1 17
45
39 21
24
r
Q 11 20
56
21
1 26 -48
1 34
33
1 16
48
37 49
23
8
12 56
57
38
1 27 28
1 34
22
1 15
47
36 15
22
9
14 33
58
56
1 28 6
1 34
9
1 14
44
34 40
21
10
11
16 10
1
13
1 28 43
1 33
53
1 13
41
33 5
20
17 47
1 1
29
1 29 17
1 33
37
1 12
37
31 31
19
1?
19- 23
1 2
43
1 29 51
1 33
20
1 11
33
29 54
18
13
20 59
1 3
56
1 30 22
1 33
1 10
26
28 18
17
14
22 35
1 5
8
1 30 50
1 32
38
1 9
17
26 40
16
15
16
24 10
1 6
18
131 19
1 32
14
1 8
8
25 3
1*
14
25 45
1 7
27
1 31 45
1 31
50
1 6
58
23 23
17
27 19
1 8
36
1 32 12
1 31
23
1 5
46
21 45
13
18
28 52
1 9
42
1 32 34
1 30
55
1 4
32
020 7
12
19
30 25
1 10
49
1 32 57
1 30
25
1 3
19
18 28
11
20
21
31 57
1 11
54 1 33 17
1 29
54
1 2
1
16 48
10
9
33 29
1 12
58 1 33 36
1 29
20
1
45
15 8
22
35 2
1 14
1
133 52
1 28
45
59
26
13 28
8
23
36 32
1 15
1
1 34 6
1 28
9
58
7
11 48
7
24
38 1
1 16
1 34 18
1 27
30
56
45
10 7
6
25
26
39 29
1 16
59
1 34 30
1 26
50
55
23
8 20
5
4
40 59
1 17
57
1 34 40
1 26
27
54
r
6 44
27
42 26 1 18
52
1 34 48
1 25
5
52
37
5 3
3
< 28
43 54
1 19
47
1 34 54
1 24
3$
5i
12
3 21
2
29
45 19
1 20
40
1 34 58
2 23
52
49
45
1 40
1
30
O
47 45
1 21
32
1 1 35 1
1 23
4
48
19
_0
a
i 1
1
9
8
7
6
to
TO
St^ns.
Sinrns.
Sifrns. * Sien*. Sign
s. Signs.
AUU
Sfrl
LOGARITHMICK
i AiiLli IX.. iVitf aecond Jiquuuuii of' urn mean la me trht"
Syzygy.
Argument. Moon's Equated Anornaiy.
Add
O
o
ft
a
en
Signs.
Sign.
Signs.
3
Signs.
4
Signs.
5
Signs.
O
eg
ci
H. M. S.
rf. M. S.
H. M. S.
H.M.S.
H.M.
S
H. M. S.
J»
o u
5 12 4tf! 8 47 8
9 46 44| 8 8
59
4 34 33
30
J
5
6
7
" 8
9
10
11
12
13
14
15
♦ 16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
C
c
10 d6
21 56
32 54
42 52
54 50
5 21 50
5 30 57
5 39 51
5 48 37
5 57 17
8 51 45
8 56 10
9 25
9 4 31
9 8 25
9 45 3
9 45 12
9 44 11
9 42 59
9 41 36
8 3
7 57
7 51
7 45
7 39
12]
23
33
46
46
4 26 1
4 17 25
4 8 47
4 7
3 .1 23
29
28
27
26
25
24
23
22
21
20
19
18
17
16
IS
14
ia
12
li
10
9
8T
7
6
5
4
3
2
1
1 5 48| 6 5 oi
1 16 46 6 14 19
1 27 44 6 22 41
1 38 401 6 30 57
1 49 33| 6 39 4
9 12 9
9 15 43
9 19 5
9 22 14
9 25 12
9 40 3
9 38 19
9 36 24
9 34 18
9 32 1
7 33
7 27
7 21
7 14
7 7
36
22
o
30
50
3 42 32
3 33 38
2 24 42
3 15 44
3 6, 45
2 2o
2 11 10
2 21 54
2 32 34
2 43 9
6 47
6 54 46
7 2 24
7 9 52
7 17 9
9 2/ 58
9 30 32
9 32 58
9 35 12
9 37 14
9 29 33
9 26 54
9 24 4
9 21 3
9 17 51
7 1
6 54
6 47
6 40
6 32
2
8
9
6
56
2 57 43
2 48 39
2 39 34
2 30 28
2 21 19
2 53 3d
3 4 3
3 14 24
3 24 42
3 34 58
7 24 10
7 31 18
7 38 9
7 44 51
7 51 24
9 39 8
9 40 51
•9 42 21
9 43 42
9 44 53
9 14 28
9 10 54
9 7 9
9 3 13
8 58 6
6 25
6 18
6 10
6 3
5 55
40
18
49
16
38
2 tit 8
2 2 53
1 53 36
1 44 16
1 34 54
3 45 11
3 55 21
4 5 26
4 25 26
4 25 20
7 57 45
8 3 56
8 9 57
8 15 46
8 21 24
9 45 '52
9 46 38
9 47 13
9 47 36
9 47 49
8 54 50
8 50 24
8 45 48
8 41 2
8 36 6
5 47
5 40
5 32
5 24
5 16
54
4
9
9
5
1 25 31
1 16 7
1 6 41
57 13
47 44
4 35 6
4 44 42
4 54 11
5 3 33
5 12 48
8 €6 53
8 32 11
8 37 19
8 42 18
8 47 8
9 47 54
9 47 46
9 47 32
9 47 14
9 46 44
9
Signs.
8 31
8 25 44
8 20 18
8 14 33
8 8 59
5 7
,4 59
4 51
4 43
4 34
56
42
15
2
33
s.
38 13
28 41
19 8
9 34
•1 I
S ; gns.
10
Signs.
8
Signs.
7
Sign
6 J "a
Signs. 1 «
Sub
tract
ARITHMETICK.
169:
LIABLE X. the third Equa-
TABLE XL The fourth Aqua-
,
tion of the mean to the true
tion of the mean to the true
Syzygy.
Syzygy; t
Argument. Sun's Anomaly.
Argument. Sun's mean distance
Moon's Anomaly
from the Node.
o
a>
n
••*
Signs. |Signs. (Signs.
a
79
/■via
Sub.
6 Add
1 Sub.
7 Add
2 Sub.
8 Add
"o
}{sig.|!{.Si B |^S«K.|0
.. '. - ' -- - ; i era
CO
6
M. S. |M- S.
M. S.
To
i\l. S. | iYl. ^. 1 ivl. 5.
30
2 22
4 12
1 22
1 22
1
5
2 26
4 16
29
1
4
1 23
1 21
29
2
10
2 30
4 18
28
2
7
1 24
1 20
28
3
15
2 34
4 21
27
3
10
1 25
1 18
27
4
20
2 38
4 24
2b
4
13
1 26
1 16
26
5|
25
2 42
4 27
25
5
6
16
1 27
1 14
25
24
61 30
T 46
4 30
24
20
1 28
1 12
7 35
2 50
4 32
23
7
23
1 29
1 10
23
8
40
2 54
4 34
22
8
26
1 SO
1 8
22
9
45
2 58
4 36
21
9
29
1 31
1 6
21
10
11
50
3 2
4 38
20
10
11
32
1 32 1 3
20
19
65
3 6 1
t 4 40
19
35
1 33 | 1
12
1
3 10
4 42
18
12
38
1 33 57
18
13
1 5
3 14
4 44
17
13
41
1 34
54
17
14
1 10
3 18
4 46
16
14
44
1 34
51
16
15
16
1 15 | 3 22
4 48
15
15
47
1 34
49 1 15
1 20
3 26
4 50
14
16
50
1 34
45
14
17
1 25
3 30
4 51
13
17
52
1 34
41
13
18
1 30
3 34
4 52
12
IB
54
1 34
37
12
19
1 35
3 38
4 53
11
19
57 1 33
34
11
1 20
^21
|'l 40
3 42
4 54
10
20
"21
1 1 1 33
31
10
i 1 45 1 3 45
4 55
9
1 2
1 32
28
9
22 1 49 1 3 48
4 56
8
22
1 5
1 31
25
8
23 1 52 3 51
4 57
7
23
1 8
1 30
22
7
24 1 56 3 54
4 57
6
24
1 10
1 29
19
6
25 2 j 3 57 4 57
5
25
1 12
1 28
16
5
26! 2 4 i 4
4 58
A
F26
1 14
1 27
13
4
27 2 9 4 3
4 58
t
27
r 1 16
1 26
10
3
28 2 13 4 6
4 58
c
> 2S
I 1 18
1 25
6
2
29 2 18 1 4 9
4 58
\
^ 2<
) 1 20
1 24
3
1
O
'2
10
) 2 22 1 4 12
4 58
(
)| 3(
) 1 22
1 22
Signs. [Signs. (Signs
?
:0
O'
CO
>^ si
M*
M**
o
en-
I
5 Suu.l 4 Sue
. 3 Sub
d9 Ad
11 Ad
djlOAd
Subtrac
t
1 2
CO
't
i
144
LOGARITHMICK
a TABLE XII. The Sun's mean Longitude* Motion* and
1
Anomaly : 9ld Style.
ft
Years
beginning
Sun's mean
Longitude.
Sun's mean'
Anomaly, j
o
2 <
■2.5
r*
CD
Sun's mean
Motion.
Suns mean
Anomaly.
s o ' "
s o ' 1
s
o ' "
so '
l
9 7 53 10
6 28 48
19
11
29 24 16
11 29 4
201
9 9 23 50
6 26 57
20
9 4
11 29 48
301
9 10 9 10
6 26 I
40
18 8
11 29 37
401
9 10 54 30
6 25 5
60
>
27 12
11 29 26
501
9 11 39 50
6 24 9
80
36 16
1 1 29 15
1001
9 15 26 SO
6 19 32
100
45 20
11 29 4
1101
9 16 11 50
6 18 36
200
1 20 40
11 29 8
1201
9 16 57 10
6 17 40
300
2 16
il 27 12
1301
9 17 42 30
6 16 44
400
9
3 1 20
11 26 16 \
1401
9 18 27 50
6 15 49
500
3 46 40
11 25 21
1501
9 19 13 10
6 14 53
600
4 32
11 24 25
1601
9 19 58 30
6 13 57
700
5 17 20
11 23 29
1701
9 20 43 50
6 13 1
800
6 2 40
11 22 33
1801
9 21 29 10
6 12 6
900
1000
6 48
7 33 20
11 21 37
11 20 11
•as
Sun's Mean
Motion.
Sun's mean
Anomaly.
2000
3000
4000
5000
6000
1
1
1
15 6 40
22 40
13 20
7 46 40
15 20
11 11 22
11 2 3
10 22 44
10 13 25
10 4 6
s o ' "
so'
1
2
3
4
11 29 45 40
11 29 31 20
11 29 17
1 49
1 1 29 45
11 29 29
11 29 14
11 29 58
11 29 42
2
o
a
*-*
cr
wo
Sun*s mean
Motion.
Suns mean
Anomaly.
5
11 29 47 29
6
7
11 29 33 9
11 29 18 49
11 29 27
11 29 11
s
o ' "
so'
Jan.
O
8
3 38
11 29 5
Feb.
i
33 18
1 33
9
11 29 49 18
11 29 40
Mar.
l
28 9 11
1 28 9
io
11 29 34 58
11 29 24
Apr.
2
28 42 30
2 28 42
11
11 29 20 38
11 29 9
May
3
28 16 40
3 28 17
12
5 26
11 29 53
June
4
28 49 58
28 24/ 8
4 28 50
13
11 29 51 7
rl 29 37
July
5
5 28 24
14
11 29 36 47
11 29 22
Aug
6
29 57 26
6 28 57
15
11 29 22 27
11 29 7
Sepi
7
29 30 44
7 29 30
16
7 15
11 29 50
Oct.
8
29 4 54
8 29 4
17
11 29 52 55
11 29 35
Nov
9
29 38 12
9 29 37
■jr is
12 29 38 35
11 29 20
Dec
10
29 12 *>2
10 29 11
AitlTHMfcTICK.
165
TABLE
XII.
concluded.
;
j S«i
n's
mean i Sun's
mean's.
mean Sun's
mean
S. mean
Motion and
Motion and
Dist. fro. Motion and
distance
Anomaly.
Anomaly.
Node
Anomaly.
fro.Node.
H 10
m|/
tt
'
H «
/
1
II
s
o
i It
a
ttl. 1
If It ■-
fi
'"M
r
11
tn\ ' //
III
1
2
U^ 59
1 58 17
S "
"~io
2
2i
rr
\
iff
ini
•■ 1
5
Ti
in
It ft ". H>
It',
2
36
1
16
231 20
30
3
2
57 25
20
4
560
5
12i 30
1
18
5lll 23
6
4
3
56 33
30
5
240
7
48j 33
1
21
19,1 25
42
5
4
55 42
40
510
10
23 34
1
23
471 28
151 30
18
6
5
54 50
50
12
190
12
50
35
1
26
54
7
6
53 58
60
14
470
15
35
36
1
28
421 33
29
8
7
53 7
7!0
17
150
18
11
37
1
31
io'i 36
5
9
8
52 15
80
19
430
20
47
38
1
33
38'1 38
40
10
9
51 23
90
22
110
23
23
39,1
36
61 41
16
11
10
50 32
100
24
380
25
58
40!1
38
34'1 43
52
12
11
49 40
110
27
60
28
34
411
41
21 46
28
13
12
48 48
120
29
340
31
10
42'1
43
30ll 49
4
14
13
47 57
13.0
32
20
33
45
43 : 1
45
57J1 51
39
15
14
47 5
HO
34
360
36
21
44 j l
48
251 54
15
16
35
46 13
15*0
36
580
38
57
45; 1
50
531 55
51
17
16
45 22
16 !
39
260
41
33
46,1
53
211 59
27
18
17
44 30
170
41
53;0
44
8
47 1
55
49 2 2
3
19
18
43 38
180
44
21Q
46
44
481
58
172 4
39
20
19
42 47
19.0
46
49,0
49
!20
49,2
442 7
13
1 21
20
41 55
20:0
49
17>0
51
56
502
3
12,2 9
50
22
21
41 3
21|0
51
45*0
54
32
512
5
40!2 12
25
23
22
40 12
22|0
54
130
57
8
522
.8
8'2 15
2
24
23
39 20
23J0
56
40.0
69
43
532
io
36
2 17
38
25
24
38 28
24,0
59
8 ! 1
i
19
&2
13
4
2 20
14
26
25
37 37
25J1
1
361
4
55
552
15
32
2 22
50
27
26
36 45
261
4
41
7
31
56 2
17
59
2 25
26
28
27
35 53
271
6
321
10
7
57i3
20
27
2 28
8
29
28
35 2
28;i
9
01
12
43
582
22
55
2 3o
32
30
29
34 10
29<f
11
28'1
15
19
592
25
23
2 33
14
31' 1
30
33 18
301
13
551
17
55
602
27
51
2 35
50
In leap years, alter February, add one day, au<
1 oho clay's motion.
22
1
166
LOGARITHMICK
i'AbLli XI 11. liquation of the Sun's Centre, or the Differ-
ence between his mean and true Place,
Argument. Sun's mean Anomaly.
Subtract.
^j
O
t i a
3
4
5 O
>3 £
r
rt
Signs
9
Sign.
Signs.
o * *
Signs.
Signs.
Signs.
o
-s •
n
o
C0
30
29
o '
' "
' "
"1
t
56 47, 1 39 6
1 55 37| 1 41 12
58 53
1|
1
59
58 30 1 40 7
1 55 39
1 40 12| 57 7
2; 3
57
1 12, 1 41 6
1 55 38
1 39 10 55 19
28
3
5
56
1 1 53 1 42 3
1 55 36
1 33 6 53 30
27
4
7
54
1 3 33 1 42 59
1 55 31
1 37
51 40
26
5
6
9
52
1 5 12j 1 43 52 1 55 24
1 35 52
49 49
25
24
u
50
1 6 50 1 44 44J 1 55 15
1 34 43
47 57
7
13
48
1 8 27i 1 45 34
1 55 S
1 33 32
46 5
23
8
15
46
1 10 2| 1 46 22
r 54 50
1 32 19
44 11
22
9
17
43
1 11 36| 1 47 8
1 54 35
1 31 4
42 16
21
10 19
40
1 13 9| 1 47 53
I 54 \7
1 29 47
40 21
20
19
11
21
37
1 14 411 1 48 36
1 53 57
1 23 29
38 25
12
23
oo
1 16 11 1 49 15
1 53 36
1 27 9
36 28
18
13
25
29
1 17 40 1 49 54
1 53 12
1 25 48
34 30
17
14
27
25
1 19 8 1 50 30
1 32 46
1 24 25
32 32
16
.15
16
29
31
20
15
1 20 34) 1 51 5
1 52 18
1 23
30 33
15
14
1 21 59
1 51 37
1 51 48 1 21 34
28 33
17
33
9
1 23 22
1 52 8
1 51 15
1 20 a
26 33
13
18
35
2
1 24 44
1 52 36
1 50 41
1 18 36
24 33
12
19
36
55
1 26 5
1 53 8
1 50 5
1 17 $
► 22 32
11
20
38
47
1 27 24
1 53 27
1 49 26 1 15 3^
20 30
10
9
21
40
39
1 28 41
1 53 5Q
1 48 46
1 13 5S
18 28
22
42
30
1 29 57
1 54 10
1 48 3
1 12 24
• 16 26
8
23
44
20
1 31 11
1 54 28
1 47 19
1 10 47
14 24
7
24
46
9
1 32 25
1 54 44
1 46 32
19 9
12 21
6
23
26
47
57
1 33 35
1 54 58 1 45 44
1 7 29
10 18
5
4
> 49
45 1 34 45
i 55 10| 1 44 53| I 5 4S
> 8 14
( 27
' 51
3?
1 35 53
1 55- 20
1 44 11 1 4 7
6 11
3
28
53
18
1 36 59
1 55 28
1 43 7 1 2 24
4 7
2
2S
55
3
1 38 3
1 55 34
1 42 10 1 1 39
2 4
1
30
56
47
1 39 6
1 55 37
1 41 12
58 55
O
1 1
iw
9
8
7
6
09
Si.rns.
S»Jrr.s.
Signs.
Si >ns.
Signs.
Signs.
•7Q
Auu
ARITHMETICK
16/
r ABLE XIV. The
Hun's
TABLE XV. Equation oj
'the &nnu
Declination.
\ie
mean Distance from the
Node.
Argument.
Sun's
tr
Argument, aun J s> meun Auomaiy.
Place.
Sis
rib
Si^ns
Signs
-
Su
tbtract.
Cf5
vV.
1
j\\
2
j\r.
n
I
2
5
4 1 5
O
6
s
7
S.
8
"~o"
2(T
20
»
Ti
24
6
? 1
30
29
CO
Sip:
Si*
Sl£.
Si
'A*
S
>£•
Si*.
IT
>
. 3
o '
O '
'
i°
'
*
11
11
30
51
"o'o
01
2;1
2 ;
4
1
4/2
5!1
50
1 4
30
1
24
"10
1
48 2
5!,1
48
1 2
29
2
48
12
11
20
36
23
20
41
61
49
2
51
47
1
28
3
l
12
12
32
20
48
27
3
6!l
81
50
2
51
46
58
27
4
l
36
12
53
20
59
26
40
9
l
101
51
2
51
45
56
26
5
l
59
13
13
13
33
21
10
25
94
5
6
11
l
121
52
2
5;1
44
54
25
24
6
2
23
21
21
13
l
141
53
2
51
43
52
7
2
47
13
53
21
31
23
7
1^1
161
54
2
4'1
41
50
23
8
3
11
14
12
21
41
9?
80
171
171
55
2
4!l
40
48
22
9
3
3414
31
21
50
91
p0
191
181
56
2
4 \ l
39
46
21
10 3
58
14
50
21
59
20
19
10 o
Tio
211
231
19
'21
1
57
2
4|7
37
44
20
11
4
22
15
9
22
8
1
85
2
3jl
3b
042
19
12
4
45
15
28
22
16
18
120
251
221
58
2
31
34
40
18
13
5
9
15
1
22
24
17
13|0
281
3011
241
59
2
31
33
37
17
14
5
3216
22
3X
16
14°
26|2
2
21
31
35
16
15 5
35
16
22
22
38
15
14
15
! 16
321
27l2
2
2|1
30
33
15
14
16 6
18
16
39
22
45
34
l
28j2
1
2 '
1]1
28
31
17
6
41
16
57
22
51
T3
' 17
| 18 j
36
1
■30J2
1
2
11
27
29
13
18
7
4
17
14
22
56
12
381
31
2
2
2
01
25
27
12
19
7
27
17
30
23
2
11
19!0
401
34
2
2
2
01
24
24
11
20
7
50
17
46
23
6
11
10
! 20
! 21
'0
42J1
35
2
3
1
59|1
23
22
10
9
21
8
15
18
2
23
.0
44 ! 1
36
2
3
1
59:1
21
20
22
8
35
18
18[23
14
8
! 920
46' 1
37
2
41
581
19
18
8
23
9
57
18
33i23
18
7
I 23
48| 1
39
2
4jl
57! 1
17
16
7
24
9
20
18
48*23
21
6
1 24'0
50|1
40
2
41
561
15
13
6
25
26
9
42
19
3 ! ,23
21
5
4
1 25',0
i ™P
52
54
11
41
2
4|l
551
13
11
5
"4
10
4
19
1723
25
;1
43J2
51
541
11
9
27
10
25119
3123
27
3
27
28!0
561
442
5
1
531
9
7
3
28
10.
47|19
4523
28
2
58 ! 1
45;2
5
1
521
8
5
2
29
11
819
58|23.
29
1
2911
o;i
462
5
1
511
6
3
1
30
a
11
30 20
Ilj23
29
30 1
2 ! 1
47!2
5
1
50 1
4
0! Q
Signs)
Si
tf.i&
Si|
-lib
a
l 1
10 1
9
8 1
7
6
-5
o
re
Q
*j
l£. J S
1 S
iff-' *"
ix r
M »•
\\
5
10
5
9
&'
3 i
5 5
jv. 4 j\r. 3
j\r
<*5
»
Add
, ^
16$
LOGARITHMICK
TAliLK XVI.
TAbLE XV11. The iuuon'tt horizontal Parallax
The Aloon's
with the Semidiametera
and true Horary Motion
Latitude in
of the Sun and Moon y and every sixth Degree
q/
£ctifl8f8.
their mean Anomalies y the Quuntitits for the in*
ter mediate Degrees being easily proportioned by
Argument.
-Moon's equa-
ted Distance
sight.
'
from Node.
S8>
g 09 |
=> c 3
* a £.
2 -
T3 3"
sT Q
*L N
• P
c
• o
UB*
S3 I
JS* (A*
i*
SI*
3
2K2
s s s
2 ~» ^
s s
O -1
P § I
D
Sighs.
Aforth ascend.
South UeoCtna
V
u
s o
/
/'
-/ "
/
//
f ~
ft
'■>'*
" js
o
1
Q
5 15
30
9,9
US*
29
15 50
14
5430
10
2
2312
2
10 30
?«
654
31 15 50
14
55'30
12
2
23
24
3
15 45
27
12*54
34115 5014
5630
15
2
23
18
4
20 5S
26
18,54
40
15 51
14
57130
19
2
23
12
5
6
26 13
31 26
25
24
24 (54
47
15 51
14
5830
59)30
26
34
2
23
6
I 0,54
56
15 52
14
2
24
11*
7
36 39
23
655
6
15 53
15
1'30
44
2
24
24
8
41 51
22
12&
17
15 54
15
430
55
2
24
11
9
47 22
21
18 55
29
15 54
15
8'31
9
2
24
12
10
11
52 13
57 23
20
19
24 55
42
15
15
12
31
31
23
40
3
25
6
2 055
56
15 58
15
17
2
25io
12
1 2 31
18
656
12
15 59
15
22 31
56
2
26
24
13
1 7 38
Vi
12 56
29
16 1 15
26*32
17
2
27
18
14
1 12 44
16
1856
48
16 21$
3032
39
2
27
12
16
16
1 17 49
1 99 *9
15
14
13
2457
8
16 415
36
33
33
11
23
6
28
6
17tl 27 53
3 057
30
16 615
41
2
28
9
16 1 32 52
12
657
52
16 815
4633
47
2
29
24
1911 37 49
11
12 58
1858
24 58
12
31
49
16 10 15
16 11 15
16 1316
52'34
5834
334
11
34
58
2
2
2
29
29
30
18
12
6
5 Signs.
A r orth dene end
4 • C
t
59
59
6
21
16 14 16
16 15 16
9'35
1435
22
45
2 '
2
30
31
8
24
\ I Signs.
South J trend
12
18
24
59
59
60
35
48
16 17 16
19 36
2436
28|36"
2
31
32
32
18
12
6
TGis 1 able
shews the
16 2C
lb* 21
10
16
20
40
2
2
!5
b0
11
16
3l;37
O
2
32
7
Moon's Lati-
6
GO
21
16 21 16
32*37
10
2
33
24
tude a little
12
60
30116 2216
37
[37
19
2
33
18
»beyond the ut-
18
60
38 116 22 16
38
37
28
2
33
12
most limits of
24
60
45 16 2316
39
37
x36
2
33
6
Ec
iipses.
|o 0,60
45 46 23,10
39
U7
40 U
331 b
ARITHMETIC*.
169
v
TABLE XVIII.
2 he Aioon'* Mean ±,ottgitude } and
Anumuiy
for current yearn.
AD. | Mean Lontr. | Mean Anom.
^unlromNo e
Years
current.| s
o ' " | s V
i n
s o
i it
Vol
7
1 8 8, 10 12
34 50
2 7
33 33
1781
H
14 42 54 11 22
19 18
1 10
43 18
1791
7
14 54 59 6 5
39 35
6 27
19 46
B
1792
7 28 40 9 17
26 44
6 7
56 52
1793
4
16 51 45 16
9 59
5 18
37 9
J794
8
26 14 51
3 14
53 14
4 29
17 26
1795
1
5 37 57
6 13
36 29
4 9
57 43
B
1796
5
28 11 37
9 25
23 38
3 20
34 49
1797
10
7 34 43
24
6 53
3 1
15 6
1798
2
16 57 48
3 22
50 8
3 11
55 23
1799
6
26 20 54
6 21
33 23
1 22
35 40
B
1800
11
5 44
9 20
16 38
1 3
15 57
1801
3
15 7 5
18
59 52
13
56 14
1802
7
24 30 11
3* 17
43 7
11 24
36 31
1803
3 53 16
6 16
26 22
11 5
16 48
B
1804
4
26 26 57
9 28
13 31
10 15
53 54
1805
9
5 50 2
26
56 46
9 26
34 11
1806
1
15 13 8
3 25
40 1
9 7
14 28
1807
5
24 36 14
6 24
23 16
8 17
54 45
B
1808
10
17 9 54
10 6
10 25
7 28
31 51
1809
2
26 33
1 4
53 40
7 9
12 8
.1
1810
7
5 56 5
4 3
36 55
6 19
52 25
1811
11
15 19 11
7 2
20 9
6
32 42
B
1812
4
7 52 52
10 14
7 18
5 11
9 48 .
♦
1813
8
17 15 57
1 12
50 33
4 21
50 5
1814
26 39 3
4 11
33 48
4 2
30 22
1815
5
6 2 8
7 10
17 3
3 13
10 39
i
B
1816
9
28 35 49
10 22
4 12
2 23
47 45
1817
o
7 58 55
1 20
47 27
2 4
28 2
1818
6
17 22
4 19
30 42
1 15
8 19
1819
10
26 45 6
7 18
13 57
25
48 36
B
1820
3
19 18 47
11
1 6
6
25 42
,
1821
7
28. 41 54
1 28
44 21
11 17
5 59
1841
12 16 37' 3 8
28 51
10 20
15 44
170
LOGARITHMICK
TABLE
xix 1 .
The Hun
'• A
.ongitudt for every day
in the
year,
at noon.
C
to
January.
February.
March.
Apri
i. 1
s
May.
o '
s
June.
s
o
t
s
o
t
s
o
i
8
o
o
"IT"
r
~2
9
11
2110
12
5411
11
80
11
551
11 12 2
2
9
12
23*10
13
5511
12
80
12
54'1
12 10 2
12
3
9
13
2410
14
5611
13
80
13
53|l
13 92
12
57
4
9
14
2510
15
57|ll
14
8,0
14
521
14 7
2
13
54
5
9
15
2610
16
57|11
15
8
15
16
51
50
1
15 5
2
14
52
6
9
16
27|10
17
5811
16
8
1
16 3
2
15
49
7
9
17
29' 10
18
5911
17
80
17
491
17 1
2
16
46
8
9
18
3010
20
Oil
18
80
18
481
17 59
2
17
44
9
9
19
'3110
21
Oil
19
■ 80
19
471
18 56
2
18
41
10
11
9
20
3210
22
1
11
20
«!o
20
45
1
19 54
2
19
39
9
21
3310
23
1
11
21
7j0
21
44
1
20 52
2
20
35
12
9
22
3410
24
211
22
7!o
22
431
21 50
2
21
33
13
9
23
3510
25
3'll
23
7,0
23
41
1
22 48
2
22
30
14
9
24
3610
26
311
24
60
24
40
1
23 45
2
23
28
15
16
9
25
37(10
27
4|11
25
60
25
39
1
24 43
2
24
25
9
26
39.10
28
4'ii
26
60
26
37
1
25 41
2
25
22
17
9
27
3910
29
411
27
50
27
36
1
26 39
2
26
19
18
9
28
4llll
5J11
28
50
28
34
1
27 36
2
27
17
19
9
29
42! 11
1
511
29
40
29
33
1
28 34
2
28
14
20
21
10
43J11
2
6;
4'l
31
1
29 32
2
29
11
10
1
4411
3
6
1
31
I
30
2
29
3
8
22
10
2
45J11
4
6
2
31
2
28
2
1 27
3
1
6
23
10
3
4611
5
7
3
21
3
26
2
2 25
3
2
3
24 J 10
4
4711
6
7
4
1*1
4
25
2
3 22
3
3
25|10
5
48;li
7
7
5
1
1
5
23
2
4 20
2
3
57
26I(T
6
49
11
8
7
6
1
6
21
2
5 17
3
4
55
27,10
7
50
11
9
8
6
59
1
7
20
2
6 15
3
5
52
2810
8
51
11
10
8
7
59
1
8
18
2
7 12
f?
6
49
29110
9
52
8
58
1
9
16
2
8 10
o
7
46
3010
10
52
9
57
1
10
14
2
9 7
3
8
43
3111
11
53
10
56
2
10 5
. .
ARITHMETICK.
171
TABLE XIX. Concluded.
July
. lAugust.
'so'
{Sept.
s o '
Octobc
IV.
r
Nov.
/
Dec.
s
o
s
o
9
i
o '
1
3
9
41
4
9
17
5
9 7;6
tf
26.7
9
15j8
9 32
2
3
10
38
4
10
14
5
10 66
9
25,7
10
158
10 33
3
3
11
35
4
11
11
5
11 4'6
10
24J7
11
168
11 34
4
3
12
32
4
12
9
5
12 2'6
11
247
12
168
12 35
5
6
3_
3
13
14
30
4
13
6
5
13 0\6
12
23;
7
7
13
14
168
16 8
13 36
14 37
27
4
14
4
5
13 596
13
22
7
3
15
28
4
15
2
5
14 57
6
14
2l!7
15
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1 72
LOGARITHMIC*
TABLE XX. J /concise
Equation-Table, adapted to the
Second Year <
ifter JLeafi'Yearj and which will be within a
Minute of the
Truth for
every Year ; shewing, to the nearest
full Minute , how much a
Clock should be faster or slower
than the Sun.
By Mr
. Smeaton.
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This Table is near enough the truth for regulating common
clocks and watches. It may be easily copied by the pen, and
being doubled, may be put into a pocket-book.
ARITHMETIC^. 149
PRECEPTS RELATIVE TO THE PRECEDING
TABLES.
To calculate the true time of New or Full Moon.
Precept 1. Write out the mean time of New Moon in March
for the proposed year, from Table I. old style, or from Table II.
in the new ; together with the mean anomalies of the Sun and
Moon, and the Sun's mean distance from the Moon's ascending
node. If you want the time of Full Moon in Marchy add the half
lunation at the foot of Table III. with its anomalies, Sec to the
former numbers, if the New Moon fulls before the 15th of
March ; but if it falls after, subtract the half lunation, with
anomalies, Sec. belonging to it, from the former numbers, and
write down the respective sums or remainders.
2. In these additions or subtractions, observe, that 60 seconds
make a minute, 60 minutes make a degree, 30 degrees make a
sign, and 12 signs make a circle. When you exceed 12 signs
in addition, reject 12, and set down the remainder. When the
number of signs to be subtracted is greater than the number
you subtract from, add 12 signs to the lesser number, and then
you will have a remainder to set down.
In the Tables, signs are marked thus, S degrees thus, ° min-
utes thus, ' and seconds thus, "
3. When the required New or Full Moon is in any given
month after Marchy write out as many lunations, with their
anomalies, and the Sun's distance from the node, from Table
III. as the given month is after March ; setting them in order
below the numbers taken out for March.
4. Add all these together, and they will give the meantime of
the required New or Full Moon, with the mean anomalies and
Sub's mean distance from the ascending node, which are the
arguments for finding the proper equations.
5. With the number of days added together, enter Table IV.
under the given month, and against that number you. have the
day of mean New or Full Moon in the left-hand column, which
set before the hours, minutes, and seconds, already found.
But (as it will sometimes happen) if the said number of days
fall short of any in the column under the given month, add one
lunation and its anomalies, &c. (from Table III.) to the foresaid
sums, and then you will have a new sum of days wherewith to
enter Table IV. under the given month, where you are sure to
find it the second time, if the first falls short.
6. With the signs and degrees of the Sun's anomaly, enter
Table VII. and therewith take out the annual or first equation
for reducing the mean to the true syzygy ; taking care to make
proportions in the table for the odd minutes and seconds of the
20
f
I5# LOGARITHMICK
iraomaly, as the table gives the equation only to whole degree^
Observe, in this and every other case of finding equations,
that if the signs are at the head of the table, their degrees are
at the left hand, and are reckoned downwards ; but if the signs
are at the foot of the table, their degrees are at the right hand,
and are counted upward ; the equation being in the body of the
table, under, or over the signs, in a collateral line with the de-
grees. The titles Add or Subtract at the head or foot of the
Tables where the signs are found, shew whether the equation
is to be added to the mean time of New or Full Moon, or sub-
tracted from it. In the table for reducing the mean to the true
syzygy, the equation is to be subtracted, if the signs of the
Sun's anomaly are found at the head of the table ; but it is to be
added, if the signs are at the foot.
With the same signs and degrees of the Sun's anomaly, enter
Table VIII. and take out the equation of the Moon's mean ano-
maly ; subtract this equation from her mean anomaly, if the
signs of the Sun's anomaly be at the head of the table, but add
it if they are at the foot ; the result will be the Moon's equated
anomaly, with which enter Table IX. and take out the second
equation for reducing the mean to the true time of New or Full
Moon ; adding this equation, if the signs of the Moon's anomaly
are at the head of the table, but subtracting it if they are at the
foot, and the result will give you the mean time of the required
New or Full Moon twice equated, which will be sufficiently
near for common Almanacks. But when you want to calculate
an eclipse, the following equations must be used : thus,
8. Subtract the Moon's equated anomaly from the Sun's mean
anomaly, and with the remainder in signs and degrees, enter
Table X. and take out the third equation, applying it to the for-
mer equated time, as the titles. Add or Subtract do direct.
9. With the Sun's mean distance from the ascending node,
enter Table XL and take out the equation answering to that
argument, adding it to* or subtracting it from the former equat-
ed time, as the titles direct, and the result will give the time of
New or Full Moon, agreeing with well regulated clocks, or
watches, very near the truth. But, to make it agree with the
solar, or apparent time, apply the equation of natural days, found
in Table XX. and you will have the true time of apparent New
or Full Moon required.
The method of calculating the time of any New or Full
Moon without the limits of the 19th century, will be shown fur-
ther on. And a few examples with the precepts, will make
the whole work plain.
N. B. The Tables begin the day at noon, and reckon for-
ward from thence to the noon following. Thus,
ARITHMETIC*.
,151
July the 1 3th, at 13 hours (4 minutes 32 seconds of tabular
time, is July 14th (in common reckoning) at 14 min. 32 sec.
past 1 1 o'clock in the morning.
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against which I find the time of mean New Moon in March,
Sun's anomaly^ Moon's anomaly, and Sun's distance from Node,
to be as set down in the example. (Agreeable to Precept I.) I
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isa XeOGARITHMICK
Moon, and the Sun's mean distance from the Moon's ascending
node ; which are the arguments for finding the proper equations,
(Precept 4.)
With the signs and degrees of the Sun's anomaly, which, in
the present case, is signs 24 degrees, I enter Table VII. and
look for signs at the top of the Table and 24 degrees in the
left hand column, and find in the angle of meeting, 1 h. 40 m.6
sec. and by making proportions, in the table (for the odd 59m.45
sec. (or estimating the Sun's anomaly at 25 deg.) I obtain, for
the 1st equation, 1 h. 44 m. 1 sec. which I apply to the time of
mean New Moon, as the title Subtract at the head of the table
directs. (Precept 6-)
With the same argument, (namely, 25 deg.) I enter Table
VIII. and take out thence the equation of the Moons mean ano-
maly, which in the present instance, I find to be 39 m. 29 sec.
which I subtract for the Moon's mean anomaly, according to the
title on the top of the Table. (Precept 7.)
The result is the argument for finding the 2d equation, with
which I enter Table IX. and take out as before the next equa-
tion, applying it to the mean New Moon, as the title directs,
(precept 7.) This gives the time sufficiently exact for common
Almanacks. But when you wish to calculate an eclipse, pro-
ceed according to Precept 8. and 9.
ARITHMETIC*.
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To calculate the time of New and Full Moon in a given Year
and Month of any particular Century between the Christian
<£ra and 18th Century.
Precept 1. Find a year of the same number in the 18th Cen-
. tury with that of the year in the century proposed, and take
•out the mWi time of New Moon in March old style, for that
year, with the mean nomalies and Sun's mean distance from the
node at that time, as already taught.
2. Take as many complete centuries of years from Table VL
as, when subtracted from the aforesaid year in the 1 8th. centu-
ry 1 , ivill answer to the given year ; and take out the first mean
15$ LOGARITHMICK
New Moon and its anomalies, &c. belonging to the said centu-
ries, and set them below those taken out for March in the 18th.
century.
3. Subtract the numbers belonging to those centuries, from
those of the 18th century, and the remainders will be the mean
time and anomalies, &c. of New Moon in March, in the given
year of the century proposed.
Then work in all respects for the true time of New or Full
Moon, as shown in the above precepts and examples.
4. If the days annexed to these centuries exceed the num-
ber of days from the beginning of March taken out in the 18th.
century, add a lunation and its anomalies, 8c c. from Table III.
to the time and anomalies of New Moon in March, then pro-
ceeed in all respects as above. This circumstance happens in
examples 6*
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To calculate the true time of New or Full Moon in any giv-
en Year and Month before the Christian iEra. *
Precept 1. Find a year in the 18th century, which being ad-
ded to the given number of years before Christ) diminished by
1, shall make a number of complete centuries.-
2. Find this number of centuries in Table VI. and subtract
the time and anomalies belonging to it from those of the mean
New Moon in March, the above found year of the V&th century ;
and the remainder will denote the time and anomalies^ Sou of
24
182
LOGArtlTHMICK
the mean New Moon in March, the given year before Chris;.
Then for the true time of that New Moon, in any month of
that year, proceed in the manner taught before.
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These Tables are calculated for the meridian of London ;
but they will serve for any other place, with equal exactness,
by subtracting 4 minutes from the tabular time, for every de-
gree that the meridian of the given place is westward of Lon-
don, or adding 4 minutes for every degree that the meridian oC
the given place is eastward of London; as inexample VII.
ARITHMETIC*.
' — A b »
.£: '.3 «A*
trie's o 2
Oh «
To calculate the true time of New or Full Moon in any giv-
en year after the 19th, Century.
Precept 1. Find a year of the same number in the 18th.
century with that of the year proposed, and takeout the time
and anomalies, &c. of New Moon in March, old style, for that
year, in Table I.
2. Take so many years from Table VI. as, when added to
the above-mentioned year in the 1 8th, century, will answer tc*
the given year in which the New or Full Moon is required ; and
take out the first New Moon, with its anomalies, for these, con*-
pletc centuries. '
m
LOGARITHMICK
3* Add all these together, and then work in all respects a?
shewn above, only remember to subtract a lunation and its ano*
roalies, when the above mentioned addition carries the New
M?on beytmd the 3 1st of March ; as in the following example *
48
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In keeping by the old style, we are always feure to be right, by
adding or subtracting whole hundreds of years to or frpm any
given year in the 18th century.
But in the new style we may be very liable to make mistajs^,
on account of the leap-year's not coming in regularly 'every
ARITHMETICS
I8S
fourth year : And therefore, when we go without the limits of
the 18th century, we had best keep to the old style, and at the
end of the calculation reduce the time to the new. '• Thus, in the
22d century there will be 14 days difference between the styles;
and therefore, the true time of New Moon* in this last exam-
ple being reduced to the new style, will be the 22 d of Jujy, at
37 in. 33 sec. past 1 P. M.
To find the times of mean New and Full Moon for ever^
month in the year;
Precept %. Find the mean time of New and Full Moon in
the month of January, as already taught, and to these times add
3 mean lunation, viz, 29 D. 12 H. 44 M. 3 S. continually, rejec-
ting the days in the month wherein the mean New or Full
Moon is required^ and the results will denote the times re-
quired.
EXAMPLE,
Let it be required to find the mean New and Full Moons in
every month of the year 1797.
Mean Full Moons.
Month*, po nor
Mean New Moons.
January
12
13 18 18
} Lunation +
29
12 44 ^
February
11
2 2 21
March
12
14 46 24
April
11
3 30 27
May
10
16 14 30
June
9
4 58 33
July
8
17 42 36
August
7
6 26 39
September
5
19 10 42
October
5
7 54 45
November
3
20 38 48
Pecember
3
9 22 51
' Months.
| D. H.
January
27 7
1 Lunation
+ 29 12
February
25 20
March
27 9
April
25 21
May
25 10
June
23 23
July
23 12
August
22
September
20 13
October
20 2
November
18 15
December
18 3
M 1 S.
40 21
44 3
24 24
8 27
52 30
36 33
20 36
4 39
48 42
32 45
16 48
S\
44 54
PERMISCUO US Q UESTIOJVS* •
1 . Required the true mean time of New Moon in May y old style*
the year before Christ 585, and the Sun's distance, at that time,
i'rom the Moon's ascending node ?
Ans. True New Moon was on the 28th day, at 2 m. 31 sec.,
past IV in the evening; and the Sun's distance from node was
3 deg. 50 m. 47 sec. (Sun eclipsed.)
2. Required the time of tiue New Moon, and Sun's distanAf
from the /iode, tn July !890 A. D. old style ?
■/
*16 LOGARITHMIC*
Ana. N. M. 29tU day III o'clock 52 m. 28 sec. and Sun's disti
from node 1 1 S. 29 ° 56' 57 ". (Sun eclipsed.*)
3. Required the true time of New Moon, and at that time the
Sun's distance from the ascending node, in April, A. D. 1819,
N. S. Ans. New Moon 24th day at 46 m.21 sec. past XI. A. M.
and Sun's dist. from node 12 ° 10 ' 9 ". (Sun eclipsed, visible.)
4. Required to find the true time of opposition of the Sun and
Moon, and the simultaneous distance of the Sun from the node,
in October, A. D. 1819, N. S.
Ans. True Full Moon 3d day, at 7m. 48 sec. past III. in the
evening, and Sun's dist* from descending node 50 ' 5 " (Moon
eclipsed, total.)
5. Required the true time of conjunction of the Sun and
Moon, September, A. D. 1820, N. S.
Ans. On the 7th day, at 16 m. 55 sec. past II. P. M. (Sun
eclipsed.)
6. Required the true time of Full Moon at Boston, Long. 70 *
37 ' 15 " W. in May, A. D. 1826, N. S.
Ans. 21st day, at 29 m. 58 sec. past X in the morning. (Moon
eclipsed.)
7. Let it be required to find the true time of New Moon m
July, 1980, O. S. and how far short the Sun will be at that time
from the Moon's ascending node.
^ Ans. 29th day, at 52 m. 28 sec. past III. in the morning.
And the Sun will be only 3 *, 3 ", short of the Moon's node, N.
ascending. (Consequently, the Sun must suffer a total eclipse.)
To calculate the true Place of the Sun for any given Moment of
Time.
Precept 1. In Tabic XII. find the next lesser year in num-
ber to that in which the Sun's place is sought, and write out
his mean longitude and anomaly answering thereto : to which
add his mean motion and anomaly for the complete residue of
years, months, days, hours, minutes, and seconds, down to the
given time, and this will be the Sun's mean place and anomaly
* Note. When the 9uh is within 17 degrees of either of the
Moon's nodes at the time of New Moon, he will be eclipsed at
that time: and when he is within 12 degrees of either of the
nodes at the time of Full Moon, the Moon will be eclipsed a*
jj^at time.— See the method of calculating Eclipses.
ARITHMETICS
J W
at that tune, in the old style ;* provided the said time be in
anyfyear after the Christian -£ra.
2. Enter Table XIII. with the Sun's mean anomaly ; and ma-
king proportions for ttte odd minutes and seconds thereof, take
out the equation of the Sun's centre : which being applied to his
mean place, as the title Add or Subtract directs, will give his
true place or longitude from the vernal equinox, at the time for
which h was required.
EXAMPLE I.
Required the Sun's true place, July 13th 1748, Old Style, at
23 hours 19 minutes 58 seconds past noon ? In common reck-
on in % t July 14th, at 19 minutes 58 seconds past XI. in the fore*
noon.
To the radical year after
Christ iroi
Add complete years
Bissextile, Days
Hours
Minutes
Seconds.
C 40
I 7
July
13
23
19
58
Sun's mean place at the given time.
EquationoftheSun'scentre,subtract
Sun's true place at the same time.
Sui
i*s Long.
| Sun's Anom.
S
'
t»
S ' '»»
9
20 43
SO
6 13 1
18
8
11 29 37
11
29 18
49
P. 29 11
5
28 24
8
5 28 24
13 47
57
13 47 57
56
40
56 40
46
46
2
2
4
3 30
20
24 58 25
— 47
57
Mean Anom.
4
2 42
23
or 2 42 26
of Cancer.
* N. B. Although this Table is constructed according to the
t>ld Style, yet it will serve, with equal exactness, for the New,
T>y diminishing the day of the month in this Table by 12, for the
present age. Thus, suppose the required time to be on the
28 th day of May N. S. Instead of the numbers answering to that
day, write out those for the 16th day, Sec. But if the required
time be within the limits of the 18th century, subtract 11 days
from the given time. See Exmnflle II, on the next page.
1
Jaa
LOGARITHMIC^
EXAMPLE II.
* Required the Sun's true place, March A 1st, 1764, jfew Stytej
at 22 hours 30 minutes 25 seconds, ffcast the noon of that day ?
Add complete years
To the radical year after
Christ 1701
;6o
3
March
3 1 st day—- 1 1 = Bissextile,Day s 20
Hours 22
Minutes 30
Seconds 25
Sun's mean place at the given time.
Equations of Sun's centre, add
Sun's true place at the same time.
Sun's Long. | Sun's Anom.
S
S
9 20 43 S0\ 6 13 1 O
27 12 11 29 26
II 29 17 11 29 14 Q
1 28 9 11
20 41 55
54 13
1 14
1
10 14 36
-h l 55 31
1 28 9
20 41 5$
54 13
1 14
1
9 1 27 23
Mean Anom.
12 10
7J or 12 10
of Aries.
EX. III. Required the Suns/ true place and anomaly, July
28th, 15h. 52m. 26 sec. past noon, in the year 1980, 9. S.
Ans. *S. 18° 35 '8 "from the vernal equinox, and Sun$
anomaly, 15.7° 18' 9"-
Tofirtd the Sun** Distance from the Moon* 9 Ascending JYode>
at the time of any given JVew or Full Moon ; and consequently
to know whether there is an Eclipse at that Time, or not.
The Sun's distance from the Moon's ascending node, is the
argument for finding the Moon's fourth equation in the syzy-
gies, and therefore it is taken into all the foregoing examples
in finding the times of these phenomina.
Thus, at the mean time of New Moon in July, 1748, the Sun's
mean distance from the ascending node is 5 S. 25 ° JO ' 1 ". See
Example /.'page 175.
The descending node is opposite to the ascending one,and they
are, therefore, just six signs distant from each other.
When the Sun is within 17 degrees of either of the nodes at*
the time of New Moon, he will he eclipsed at that time : and
when he is within 12 degrees of either of the nodes at the time
ARITHMETICS.. M
of Full Moon, the Moon will be eclipsed.* Thus we find these
-will be an eclipse of the Sua at the time of New Moon in July,
1748.
But the true time of that New Moon comes out by the equa-
tions to be £ minutes 10 seconds later than the mean time
thereof, by comparing these times in the above example : and
therefore, (in this, and all similar cases) we must add the Sun's
motion from the node during that interval to the above mean
distance, 5 S. 25 ° 30 » 1 '% which motion is found in Table XII
for 6 minutes, 10 seconds, to be 14 ". And to this we must ap-
ply the equation of the Sun's mean distance from the node, in
Table XV. found by the Sun's anomaly, which at the mean time
of New Moen in example I. we estimated at 25 °, and then we
shall have the Sun'* true distance from the node, at the true
time of New Moon, as follows ;
Sun from Node,
sO'"
At the mean time of New Moon? e , ! « A ,
in July, 1748, £ 5 25 30 1
Sun's motion from the > 6 minutes \ 1 4
node for J 10 seconds. J O
Sun's mean distance from node,? _ - «. n ,-
at true New Moon $ 5 **• oU 15
Equation of mean distance from ? -«
node, subtract } ♦
Sun's true distance from < 5 25 29 23
the ascending node ; that is, { 4 • 30 ' 37 **■ from
the descending node ; which being far within the above limit
of 17 degrees, shows that the Sun must then be eclipsed..
And now we shall shew how to project this, or any other
eclipse, either of the Sun or Moon.
* Note. This admits of some variation : for inapogeal eclipses 4
the solar limit is but 1 6\ degrees ; and in perigcal eclipses, it
is 18}. When the Full Moon is in her apogee, she will be eclip-
sed if she be within lOf degrees of the node ; and when she is
full in her perigee, she will be eclipsed if sh$ be within 12^.
degrees of the node.
25
if* LOGARITHMIC*
TO PROJECT AN ECLIPSE OF THE SUN.
In order to this, we must find the ten following Elements by
means of the Tables.
1. The true time of conjunction of the Sun and Moon; and
at that time,
2. The semidianieter of the Earth's disk* as seen from
the Moon, which is always equal to the Moon's horizontal par*
*llax.
3. The Sun's distance from the solstitial colure to which he
is then nearest.
4. The Sun's declination.
5. The angle of the Moon's visible path with the ecliptick.
6. The Moon's latitude.
7. The Moon's true horary motion from the Sun.
8. The Sun's semidianieter.
9. The Moon's ,semidiameter.
10. The semidianieter of the penumbra.
Wc shall now proceed to find these elements for the Sun's
Eclipse in July, 1748, O. S.
1. To find the true time of JYew Moon. This, by example I.
page 175, is found to be on the 14th dav of the said month, at 19
minutes 58 seconds past XL in the morning.
2. To find the Moon 9 8 horizontal parallax ', or semidiameter of
the Earth'* disky as seen from the Moon. Enter Table XVIL
with the signs and degrees of the Moon's anomaly, (making
proportions, because the anomaly is in the table only to every
6th degree,) and thereby take out the Moon's horizontal paral*
lax ; which, for the above time answering to the anomaly 10 °
56 '56", is 54* 33".
3. To find the Sun's distance from the neartst solstice, viz.
the beginning of Cancer, which is 3 signs } or 90 degrees fromthe
beginning of Aries.
It appears by the example on page 187 (where the Sun's
place is calculated to the above time of New Moon) that the
*. Note. The body, or face of the Sun, or Moon, as it appears,
to a spectator on the Earth j or of the Earth, as it would appear
to a spectator at the Sun, or Moon, is called its Disk.
ARITHMETICS. in
Sun'fi longitude from the beginning of Aries is there 4S. 2° 42*
33", that is 2° 42* 23" from the beginning of Cancer ; Thus
SO'"
From the Sun's Longitude or place 4 2 42 23
Subtract, . ■ 3
Remains the Sun's distance from the > 1 2 42 23
solstice of Cancer. J
Or, 32° 42* 23" : each sign containing 30 degrees.
4. To find the Sun'* declination. Enter Table XIV. with the
signs and degrees of the Sun's true place, viz. 4£, 2° and mak-
ing proportion for the 42' 23", take out the Sun's declination
answering to his true place, and it will be found to be 19° 38' 8"
North.
5. To find the Moon's latitude. This depends on her distance
from her ascending^ node, which is the same as the Sun's dis-
tance from it at the time of New Moon : and with this the
Moon's, latitude is found in Table XVI.
Now we have already found that the Sun's equated distance
from the ascending node, at the time of New Moon in July
1748, is 5$. 25° 29* 23". See the 189th fiage. Therefore,
enter Table XVI. with 5 signs at the bottom, and 25 and 26 de-
grees at the right hand counted upward, and take out 26* 13',
the latitude for SS. 25° ; and 20 e 59', the latitude for 5&. 26' :
and by making proportion between these latitudes for the 29*
23" by which the Moon's distance exceeds the 25th. degree;
her true latitude will be found to be 23 ' 56 " North Ascending.
6. To find the Moon's true horary motion from the Sun. With
thfe Moon's anomaly, viz. OS. 10*56' 56", enter Table XVII.
*nd take out the Moon's horary motion ; which, by making
proportion in that table, will be found to be 30' 14". Then,
with the Sun's ajnomaly, 25°, take out his horary motion 2' 23"
from the same table ; and subtracting the latter from the for-
mer, there will remain 27' 51" for the Moon's true horary mo*
tion from the Sun.
7. To find the angle of the Moon's visible fiath with the Eclifi-
tick. This, in the projection of eclipses, may be always rated
at 5° 55', without any sensible errour.
8,9. To find the semi diameters of the Sun aud Moon. These
are found in the same Table, and by the same arguments, as
their horary motions.— In the present case the Sun's anomaly
gives his semidiameter 15' 51", and the Moon's anomaly gives
her diameter 14' 56'*.
10. To find the semidiameter of the Penumbra. Add the
Moon's semidiameter to the Sun's, and their sum will be the
serajdiarneter of the penumbra, viz. 30' 47".
o
»
»»
54
33
32
42
23
19
38
8
23
36
37 51
5
35
a
15
51
14
56
30 4T
t» LOGARITHMIC*
Now collect these elements, that they may be found the mom
readily when they are wanted in the construction of this Eclipse,
D. H. M. S.
1. True time of New Moon in July, 1748, 14 11 19 58
2. Sun's diameter of Earth's disk,
3. Sun's distance from the nearest solstice,
4. Sun** declination, North,
5. Moon's latitude North descending,
6. Moon's horary motion from the Sun,
7. Angle of the Moon's visible path with
the ecliptick,
8. Sun's semidiameter,
9. Moon's semidiameter,
40r Semidiameter of the penumbra,
TO PROJECT AM ECLIPSE OF THE SUM
GEO ME TRICALL F.
Make a scale ef any convenient length, as A. C. (Fig. 1.) and
divide it into 60 equal parts, reckoning each part to be one
minute, or the sixtieth part of a degree.
Then, take the semidiameter of the Earth's disk, 54 minutes,
33 seconds, (or 54$) from the scale, in your compasses ; and with
that extent, set one foot in the end C of the scale as a centre $
and with the other foot describe the semicircle A D B, for the
circumference of the northern half of the Earth's illuminated
disk, or surface, because we live on the north side of the Equa-
tor ; continue the line A C to B ; so A C B shall be a portion
of the Ecliptick, equal to the diameter of the Earth, as seen from
the Sun, or Moon at that time.
Upon the centre C, raise the straight line C D H, perpendic-
ular to A C B ; and call the line C D H, the axis of the ecliptick.
Being provided with a good sector, open it to the radius C A
in the line of chords; and taking from thence the chord of 23$
degrees in your compass, set it off both ways from D to G and
to E, in the periphery of the semi-disk. [But, as much the
greater number of those into whose hands this work may faJL,
are not supposed to be thoroughly skilled in the use of Mathe-
matical Instruments, we shall pursue somewhat a different
method ; which, in point of simplicity and precision, is no less
preferable :] Or :—
Divide the quadrants A D and D B, each into 90 equal pajts,
ARITHMETICS. l«
ifcr degrees, beginning at D. Then connect the points E and Gr
(which are distant 2 3£ degrees on each side of D) with the
straight line E F G ; in which the North pole P of the Earth's
disk will always be found.
When the Sun is in Aries, Taurus, Gemini, Cancer, Leo, and
Virgo, the North pole of the Earth is enlightened by the Sun ?
but while the Sun is in the other six signs, the South pole i»
enlightened, and the North pole is in the dark.
And when the Sun is in Capricorn, Aquarius, Pisces, Aries,
Taurus, and Gemini ; the northern half of the Earth's axis
C XII P lies to the right hand of the axis of the ecliptick, as
seen from the Sun ; and to the left hand, while the Sun is in the
other six signs.
The order, and the names of the Signs, the months and days
of the year, in which the Sun appears to enter these Signs, are
as follows :
J l >
(2-)
(30
<*•)
(«•)
(60
Arte a f
Taurus,
Gemini,
Cancer,
Leo,
Virgo,
March,
April,
May,
June,
July,
August,
20,
20,
31,
21,
23,
23,
(*•) (•■) (9.) Co,) (.11,): (12,)
Libra, ocorflio, Sagittarius, Cafiricornus, Aquarius, JPiscct,
Sept. October, November, December, January, February*
23, 23, 22, 21 20, 19.
Open the sector, till the radius (or distance of the two 90*s)
of the signs be equal to the length of D G, and take the sine of
the Sun's distance from the solstice (32 8 42 * 23 ") as nearly
as you can guess, in your compasses, from the line of sines, and
set off that distance from F P, in the line E F Q, because the
Earth's axis lies to the left hand of the axis of the ecliptick, as
«een from the Sun in the month of July. Or ;
Set one foot of the compasses in the point F, where the line
E F G intersects the axis of the ecliptick CDH; and, having
extended the other foot from F to E, or from F to G, describe
the semicircle E H G, and divide its quadrant H E into 90 equal
parts or degrees. — If the Earth's axis had lain to the right hand
from the axis of the ecliptick, the quadrant H G must have
been divided into 90 degrees, and not the quadrant HE.
As the Sun is 32 degrees 42 minutes 23 seconds, (which
may be estimated 3 2 -J- degrees) from the nearest (or summer)
solstice, which is the lirst point of Cancer, on the noon of the
14th of July 1748, draw the right line I P, parallel to H D, from
32| degrees of the quadrant H E till it meets the line E F G at
*9* LOGARITHMIC*
P, then from P to C, draw the right line PC; so P C shall he
the northern half of the Earth'* axis, and P the North, pole.
As the Sun is on the North side of the Equator in, July , and
Consequently nearer the point of the heaven just over London
(or the vertex of London) than the Equator is ; subtract his de-
clination, 19 degrees 38 minutes (neglecting the 8 seconds)
from the Latitude of London, 51 degrees 30 minutes, and the
remainder will be 31 degrees 52 minutes, for the Sun's distance
from the vertex of London on the noon of July the 14th.
From the point k (in the right hand side of the semicircle
ADB)at 31 degrees 52 minutes, counted upward from B,
draw k I, parallel to C D : and taking the extent* / in your
compasses, set it from C to XII on the Earth's axis C P. So,
the point XII shall be the place of London, at the instant when
it is noon at that place on the 14th. of July 1748.
Add the Sun's declination 19° 38% to the Latitude of London
51° 30*, and the sum will be 71 degrees 8 minutes, for the Sun's
distance from the vertex of London on the 14th of July at mid-
night. Therefore,
From 71° £', counted upward from B to m in the right hand
side of the semicircle A D B, draw the right line m n parallel
to C D.
Then, taking the extent m n in your compasses, set it from
C towards or beyond P on the Earth's axis C P, as it happens
to reach short of P or beyond it : but in the present case, it
reaches so little above P, that we may reckon C P, lo be its
whole extent : and so, the point P shall lepresent the place or
situation of London at midnight, beyond the illuminated part of
the Earth's disk, as seen from the Sun ; and consequently, in
the dark part thereof!
Divide the part of the Earth's axis between XII and P into
two equal parts, XII X and P K ; then, through the* point K,
draw the right line VI K VI (till it meets, on each side, the
periphery of the disk) perpendicular to the Earth's axis C XII
K P.
Now, to draw the parallel of latitude of any given place, as
suppose London, or the path of that place on the Earth's en*
lightened disk, as seen from the Sun, from Sun-rise till Sun-set,
proceed as follows*
Subtract the Latitude of London, 5l p 30% from £0° 00', and
there will remain 38| for its colatitude, which take in your com*
paste's, from the line of chords, making C A or CB radius 5 Or,
From 38$ degrees, counted upward from B to y in the semi-
circle A D B, draw the right line v w ; and, having taken its-
length in your compasss, set off that extent both ways from K in
the Earth's axis, to VI and VI, in the line VI K VI.. ; -,
A*UT»MET1CR. tfji
The compasses being opened from K to VI, set one foot in K as
a centre } and with the other describe the semicircle VI 7 8 9 10
i 1 12 1 2 3 4 5 VI, and divide it into 12 equal parts, Then,
from these points of division (7 8 9, Sec.) draw the dotted lines
7 a, 8 by 9 c, 10 d 9 Sec. all parallel to the Earth's axis C XII P,
as in the figure.
With the small extent P K as radius, describe the semicir-
cle P 6 5 4 3 2 1 XII, and divide the lower quadrant into 6 equal
parts as in the points 1,2, 3,4,5, 6 ; because the Sun has
North declination.
But if the Sub had South declination, the other quadrant
must have been so divided.
Through the said division points of the quadrant XII 1 2 S 4
Ice. draw the rightlinesXI 1 XI, X 2 X, IX S IX, VIII 4 VIII
VII 5 VII, all parallel to the right line VI K VI ; and through
the points where these lines meet the former parallel lines 7 a,
8 by 9 c, 10 dy &c draw the elliptical curve VI VII VIII IX X
XI XII I II III IV V VI, which may be done by hand, from
point to point ; and set the hour-letters to these points where
the right lines meet in the curve, as in the figure.*
This curve shall represent the parallel of Latitude of London,
or, the path which London (by the Earth's motion on its axis)
appears to describe on the Earth's disk, as seen from the Sun on
the 14th of Julyy from VI in the morning untill VI in the eve*
ning : and the points VI, VII, VIII, IX, Sec. in the curve shall
be the point of the disk where London would be at each of these
hours respectively, as seen from the Sun.
If the Sun's declination had been as far South as it was North,
the diurnal path of London would have been on the upper side
of the line VI K VI ; that is the ellipse, of which the curve VI
VII, VIII, IX, X, &c. is a part, would have been complete?
and must have been regulated by dividing the upper quad-
rant P 6 (of the small semicircle) into 6 equal parts, and
drawing lines parallel to VI K VI, as before, till they meet
the intercepting lines drawn through the division points of the
quadrant P C. The points in which this elliptical curve would
touch the periphery of Earth's disk, would denote the instant
of the Sun's rising, and of setting at the given place.
MakeC AorCB the radius of a line of chords on the sec*
tor, and take therefrom the chord of 5 ° 35 ', the angle of the
Moon's visible path with the Ecliptick : Or,
From the point M, viz. at 5 degrees 35 minutes, to the right
*N. B. The hour letters on the right hand side of XII, to-
wards VI (in the Figure) viz. XI X IX VIII VII, is an errour
in sculpture ; it ought to be I ll III IV V Sec. The reader is
therefore required} to correct this trivial mistake ki projecting
Eclipses.
1,96 LOGAMTHMICK
hand of the axis of the Ecliptick C Dy draw the right line
M C, for the axis of the Moon's orbit as seen from the Sun, be-
cause the Moon's Latitude is North de acending % on the 14th/uty
1748. If her Latitude had been North ascending^ the axis of
her orbit must have been drawn 5 degrees 35 minutes on the
left hand from the axis of the Ecliptick. .
N. B. The axis of the Moon's orbit lies the same way when
her Latitude is South-ascending, as when it is North-ascend*
ing ; and the same way when South descending, as when North
descending.
Take the Moon's. Latitude 23' 36", from the scale C A in
your compasses, and set that extent from C to q on the axis*
(C D) of the Ecliptick. Then, through the point y, draw the
right line IX X XI q z XII 1 &c. perpendicular to the axis of
the Moon's orbit C z M ; and this line shall be the path of the
centre of the Moon's shadow over the Earth r and will repre-
sent as much of the Moon's orbit, seen from the Sun, as she
moves through, during the time that her shadow or penumbra is
passing over the Earth-
. From C, on the scale AC, take the Moon's horary motioa
from the Sun, 27* 51 ", in your compasses ; and make the
small scale A B (Fig. 2.) equal in length to that extent .• and
divide this scale into 60 equal parts, for so many minutes of
time. Then, as the time of New Moon, on the 14th of July,
1748, was 19 minutes, 58 seconds, after XI o'clock, take 19
minutes, 58 seconds, or, in this case, 20 minutes, (not regard*
ing the 2 seconds) counted from A to « en the small scale A B
in your compasses, and set them off, (in Fig. !.) from the middle
point between q and £, in the path of the penumbra's centre, to •
XI in that line ; because the instant of tabular time of New
Moon is exactly between the point q, where the axis C D of the
Ecliptick, and the axis C M of the Moon's orbit, intersect the
line, or path of the penumbra's centre on the Earth.
Take the whole length of the scale A B (Fig. 2.) in your com-
passes ; and with that extent, make marks along the line IX X
XI XII I, Sec both ways from XI ; and set the hour-letters to
these marks, as in the figure. Then, from the scale A B (Fig.2.)
divide each space, from mark to mark, into 60 equal parts, or>
horary minutes, which shall shew the points of the Earth's disk
where the centre of the penumbra falls, at every hour and min-
ute, 1 during its transit over the Earth.
[To the Binder.— JLet the Plate face this page, and unfold to
the right.]
cttheSuxb&ty*
observed
Afnr**"*""* pis, Old StOr.
^o
A
itr-.
_30
ARITHMETICS <y>7
Apply one side of a square* to the line of the penumbra's
path, IX X XI, &c. and move the square forward and back*
ward till the other side cuts the same hour and minute, as at
r and «, both in|the path of the penumbra's centre, and the path of
London : and the minute which the square cuts at the same
instant in both, these paths, is the ins tan c of the visible conjunc-
tion of the Sun and Moon, or the greatest obscuration of the
Sun, at the place for which the construction is made, namely,
Lx>ndon f in the present example ; and this instant, according to
the projection, is at 34£ minutes past X o'clock in the morning;
Take the Sun's semidiameter 15 '51 "in your compasses, j
from the scale A C, (Fig. 1 .) j and setting one foot at r, as a cen-
tre in the path of London, namely, at 34J minutes past X, with <
the other foot describe the circle R S for the Sun, or which shall ;
represent the Sun's disk as seen from London, at the greatest j
_ obscuration. \
Then take the Moon's semidiameter 14' 56" in your com-
passes from the scale ; and setting one foot in the Moon's path ' -\
at *, 34* minutes past X, with the other foot describe the circle <
T U, for the Moon's disk, as seen from London, at the moment
when the eclipse is at the greatest ; and the portion of the Sun's
disk which is hid or obscured by the Moon's, will shew the
quantity of the eclipse at that time ; which quantity may be *
measured on a line as, 1 2 3 4 5 6, Sec. equal to the Sun's diam*
eter, and divided into 12 equal parts for digits ;t of which ac-
cording to the present projection, there are 9f digits eclipsed.
Lastly, take the semidiameter of the penumbra 30' 47" from
the scale C A, (fig I.) in your compasses ; and setting one foot
in the path of the penumbra's centre, direct the other foot t©
the path of London among the morning hours at the left hand ;
and carry that extent backward and forward, till both points of
the compasses fall into the same instant in both the path's ; and
that instant will denote the time when the Eclipse began at
London. Then, do the like on the right hand of the axis of
the ecliptick ; and where the points of the compasses fall into
the same instant in both the paths, that instant will be the time
when the Eclipse ended at London. «
These trials give 7\ minutes after IX in the morning for the
the beginning of the Eclipse : 34£ minutes after X, for the time
of greatest obscuration ; and 13* past XII, for the time when
the Eclipse ended.
Note. *The learner will find it convenient to be provided
with a small wooden square , the two sides of which are about
6 inches in length*
t A Digit is a 12th part of the apparent diameter of the
Sun or Moon.
26
m L0GARITHM1CK
From these times we must subtract the equation of natural
days, viz. 6 minutes, in July 14th. and we shall have the appar-
ent times ; nanaeJy, I minute SO seconds past IX, for the be-
ginning of the Eclipse, 28 minutes SO seconds past X, for the
time of greatest obscuration, and 7£ minutes past XII for the
time when the Eclipse ended. But the most convenient way is
to apply this equation to the true equal time of New Moon,
before the projection be begun, as is done in Example I.
For the motion or position of places on the Earth's disk, an
swer to apparent or solar time.— f See Mr. Patterson's Edition
of Fuge son's Astronomy % page 340,— also his introduction t&
Agronomy y p. 163—171, and End. Mat Phil, fi. 391—394.
TO PROJECT AM ECLIPSE OF THE M00M.
When the Moon is within 12 degrees of either of her Nodea^
at the time when she is full, she will be eclipsed, otherwise she
Will not.
We find by Example III. page 178, that at tlie time of mean
Full Moon in April) the Sun's distance from the ascending node
was 1 IS 26° 53* 2" ; that is only 3* 6' 58" short of her descen-
ding node, and the Moon being then Opposite to the Sun, must
have been just as near her ascending node, and was therefore
eclipsed.
The Elements for constructing an Eclipse of the Moon are
eight in number, as follows :
1. The true time of Full Moon .• and at that time.
2. The Moon's horizontal parallax,
3. The Sun's semidiameter.
4. The Moon's semidiameter.
5. The semidiameter of the Earth's shadow at the Moon.
6u The Moon's Latitude.
7* The angle of the Moon's visible path with the Ecliptick.
3.. The Moon's true horary motion from the Sun.
Therefore,
* 1. To find the true time of Full Moon. Work as already
taught in the Precepts. — Thus we have the true time of Full
Moon iu Aprils 1819, (see Example II L page J 78, J on the
10th day, at 1 1 minutes 48 seconds past I o'clock, P. M.
2. To find the Moon's horizontal parallax. Enter Table
XVII. with the Moon's mean anomaly (at the above time of Full
Moon) 2 5 27 ° 43 ' 47 ", and thereby take out her horizontal
parallax ; which, by making the requisite proportion, will be
found to be 57 • 20 ".
ARITHMETICS W
3, 4. To find the aemidiameter tf the Sun and Moon. Enter
Table XVII. with their respective anomalies, the Sun's being
9 5 7 ° 38 ' 53 ", (by the aforesaid Example) and the Moon*s 2
S 27 ° 43 " 47 " ; and thereby take out their respective semidi-
ameters : The Sun's 16 ' 4 ", and the Moon's 15 ' 38 ".
5. To find the a e mi diameter of the Earth* 8 shadow at the
Moon. Add the Sun's horizontal parallax, which is always 10 ",
to the Moon's, which, in the present case is 57 ' 20 ", and the
sum will be 57 * 30 ", from which subtract the Sun's semidiame-
ter 1 6 ' 4 ", and there will remain 41 * 26 M for that part of t|te
Earth's shadow which the Moon then passes through.
• 1. To find the Moon* a Latitude, Find the Sun's true dis-
tance from the ascending node (as already taught in page 189)
at the true time of Full Moon ; and this distance, increased by
six signs, will betfte Moon's true distance from the same node^
and consequently the argument for fiindingher true latitude, as
shewn in page 191.
Thus, in Example III. the Sun's mean distance from the as-
cending node, was 115 26 ° 53 ' 2 ", at the time of mean Full
Moon : but it appears by the Example, that the true time there-
of, was 13 hours, 55 minutes, 33 seconds, later than the mean
time, and therefore we must add the Sun's motion from the
node (found in Table XII.) during this interval, to the above
mean distance 1 13. 26° 53' 2", in order to have his mean dis-
tance from it at the true time of Full Moon. Then to this apply the
equation of his mean distance from the node (found in Table
XV.) by his mean anomaly 95 7° 38' 53" ; and lastly, add si*
signs s so shall the Moon's true distance from the ascending
node be found, as follows :
s ' M
Sun's distance from node at mean Full Moon, II 26 53 2
f 13 hours 32 2
Add his motion from it in< 55 minutes 2 15
1 33 seconds, 1
11
27
27
20
2
4
11
29
31
20
6
Sun's mean distance at true Full Moon*
Equatipn of his mean distance, add
Sun's true distance from the node.
To which, add
And the sum will be 5 29 31 20
Which is the Moon's true distance from her ascending node
at the true time of her being full ; and consequently the argu-
ment for finding her true latitude at that time. Therefore,
with this argument, enter Table XVI. making proportion,
trelwcen the latitudes belonging to the 5th and 6th degree of
200 LOGARITHMICK
the argument at the right hand (the signs being at the bottom)
for the 31* 20", and it will give 2' 41" for the Moon's true Lat-
itude, which appears by the Table to be North descending.
7. To find the angle of the Moon 9 a visible path with the Eclifi-'
tick. This may be reckoned 5° 35% without any perceivable
errour in the projection of Eclipses.
8. To find the Moon's true horary motion from the Sun.
With their respective anomalies take out their horary mo-
tions from Table XVII. and subtract the Sun's horary motion
from the Moon's ; the difference will be the Moon's true hora-
ry motion from the Sun : in the present case 30* 49".
, Now collect these elements together for use.
D H M S
1. True time of Full Moon in Aftril, 1819 10 1 1148
2 r . Moon's horizontal parallax, 57 20
3. Sun's semidiameter 16 4
4. Moon's semidiameter, 15 38
5. Semidiameter of Eearth's shadow at the Moon, 41 26
6. Moon's true Latitude, North descending, 2 41
7. Angle of the Moon's visible path with the > 5 #5 o
Ecliptick, 5
8. Her true horary motion from the Sun. 30 49
These Element* being accurately prepared for the construc-
tion of the Moon's Eclipse in April 1819, proceed as follows :
Make a scale of any convenient length, A B, Fig. 3. and di-
vide it into 60 equal parts, each part answering to a minute of
a degree. «
Draw the right line A B, (Fig. 4.) for part of the ecliptick*
and R D perpendicular to A B for the northern part of its axis ;
the Moon having North Latitude.
Add, the semi diameters of the Moon and Earth's shadow to-
gether, which, in this Eclipse, will make 56' 4" ; and take this
from the scale in your compasses, and setting one foot in the
point where the axis R D of the Ecliptick meets the right line
A B as a centre, describe the circle AJD E N O ; in one point
of which the Moon's centre will be at the beginning of the
Eclipse, and in anpther point opposite to the former, at the end
of the Eclipse.
N. B. If the Moon's NorthL atitude had been equal to twice
her semidiameter, it would have been sufficient to describe on-
ly the semicircle ADEN.
But in case her Latitude had been South, and equal to twice
her semidiameter ; we must have described the semicircle N«
O A. When her Latitude (whether North or South,) is less
ARITHMETICS 201
than twice her semidiameter, it will be best to describe a com-
plete circle, as in the Plate, fig. 4.
Take the semidiameter of the Earth's shadow, 41* 26'% in
your compasses from the scale, and setting one foot in the same
point for a centre as before, with the other describe the circle
W L Y M for the whole circumference of the Earth's shadow
at the Moon, through which she passes at her full, A fir it 1819.
Make R D the radius of a line of chords on the sector, and
set off the angle of the Moon's visible path with the Ecliptick*
3° 35', from D to E ; (or, by dividing the quadrant DEN into
90 equal parts, as in Fig. 1) and draw the right line T E for the
northern half of the axis of the Moon's orbit, lying to the right
hand from* the axis of the Ecliptick R D, because the Moon's lat.
is North descending. It would have been the same way (on
the south side of the ecliptick) if her Latitude had been South
descending ; but contrary in both cases, that is, to the right
hand from the axis of the Ecliptick ;■ if her Latitude had been
either North attending or South ascending.
Take the Moon's Latitude, ?* 41", from the scale, in your
compasses, and set off that extent from the point in which the
perpendicular R D falls upon the right line A W B,te T in the
axis of the Moon's orbit ; and through the point T, at
right angles toT E, draw k the right line P R T N for the path
of the Moon's centre.
Then, T shall be the point in the Earth's shadow, where the
Moon's centre is at the middle of the Eclipse ; the midle point
between R and T (which was not designated for want of room,)
will be the point where her centre is at the Tabular time of her
being full ; and R, the point where her centre is at the instant
of ecliptical opposition.
Take the Moon's horary motion from the Sun, 30 * 49 ", in
your compasses from the scale A B (Fig 3.) and with that extent
make the small scale (Fig. 5.) an divide it into 60 equal parts,
or horary minutes* — Then as the true time of Full Moon in
April 1819, was at 1 1 minutes 48 seconds, or 1 If minutes past I
o'clock; take ll£ minutes from the (last mentioned) scale in
your compasses, and set that extent from the point, signifying
the instant of Full Moon (which is mid-way between R and T)
to the left on the line (P R T N) of the Moon's centre, so shall
that extent fix the point where the centre of the Moon is at the
instant when it is I o'clock oX London.
From this pointl, with the whole length of the scale (Fig. 5.)
in your compasses, make marks along the whole length of the
line in the path of the Moon's centre, and set the hour letters to
these marks, as in the figure : then divide each space from mark
-to mark, into 60 equal parts or horary minutes, as in (Fig. 5.)
2P2 LOGARITHMIOK
Take the Moon ( 8 semidiameter, 15 * 38 ", in your compasses,
from the scale A B, and with that extent, as a radius, upon the
points N> T, and P, as centres, describe the circle Q for the
jVIoob at the beginning of the Eclipse, when she touches the
Earth's shadow at Y ; the circle R for the Moon at the middle
of the Eclipse, and the circle S for the Moon at the end of the
Eclipse, just leaving the Eauth's shadow at W.
The point N denotes the instant when the Eclipse begins,
namely, at 25 minutes 30 seconds after XI in the morning : the
point T the middle of -the Eclipse, at 10 minutes 18 seconds
past I o'clock in the afternoon ; and the point P the end of the
Eclipse, at 58 minutes after II. — Thus it appears, that the Moon
was totally eclipsed for the space of 2 hours, 42 minutes, SO
seconds.
MOnJS EXAMPLES*
Exp. Let it be required to find the Elements for the Solar
Eclipse which happened in AjiriL 1764, New Style.
D. H. M. S.
I. True New Moon Afiril, 1764. 1 10 30 25
2. Semidiameter of the Earth's disk,
3. Sun's distance from nearest solstice,
4. Sun's declination, North.
5. Moon's Latitude, North ascending.
6. Moon's horary motion from the Sun;
7. Angle of the Moon's visible ?
path with the Ecliptick. J
8. Sun's semidiameter,
S. Moon's semidiameter,
# 10. Semidiameter of the Penumbra.
This Eclipse was nearly central, and annular.*
o
9
M
54
43
77
49
53
4
49
40
18
27
54
5
35
16
6
14 57
31
S
* Note. In annular eclipses, the light of the Sun is left all
around the Moon in a circulr form. Annular, from the Latin
ctnnulusy a ring.
9
>
*>
57
20
IS
56
15
59
41
34
32
21
5
35
Q
30
52
ARITHMETICS. TO
"Exp. Let it be required to find the Elements for the Lunsp
Eclipse in May, 1762, N. S.
D, H. M, S.
1. True Full Moon in May, 1 7 62. 3 5 50 50
2. Moon's horizontal parallax,
3. Sun's semidiameter.
4. Moon's semidiameter.
5. Semidiameter of Earth's shadow?
at the Moon. y
6. Moon's true Latitude, South descending.
7. Angle of the Moon's visible >
path with the Ecliptick. }
8. Her horary motion from the Sun.
Ex. 3. Required the Elements for the Eclipse of the Sun,
^Ari/ 24th, 1819 ?
jEx. 4. Required the Elements for the Lunar Eclipse, Oct.
'3d, 1819? (total.)
Ex. 4. Let it be required to calculate the Elements for the
Lunar Eclipse, March 29th, 1820 ?
Ex. 6. In the year 1823, there will be four Eclipses ; namely,
two of the Sun, one February 11th ; and th* other July the 8th z:
and two of the Moon, one January 26th, and the other July 23d*
(both total.) Let it be required to tind the respective Ele*
ments for the construction of these Eclipses ?
Ex. 7. In the year 1826, there will be two Eclipses of the
Moon ; viz. May 2 1st, and November 1 1th, (both total.) What
are the Elements belonging to each I
Ex. 8. What are the proper Elements for constructing apt
Eclipse of the Sun, which will happen July 29th, J980, Okl
Style ? .
To find the number of Eclifises there are in any given year,
•and in what Months they hafifien.
Precept. Enter Table XVIII. and take out the mean Lon-
gitude of the Moon's Nodes for the given year ; with which en-
ter Table XIX. and find, in that table, when the Sun'-s Longw
tude will be nearly the same of six signs different ; and the day
of the month in which these numbers are so found, will be the
time required.
EXAMPLES.
1. It is required to find the number of Eclipses in the year
1796 ; and in whut months they will happen.
The mean Longitude of the Moon's* Noi th Node, on the first?
of January 1796, is S5 20° 35', of the South Node (it being just
l
204 LOGARITHMICK.
six s^gns distant) 9S 20° 35* ; wherefore the Node-mtmtbs- are
January, July , and December ; consequently there were three
Eclipses in that year.
2. Required the number of Eclipsed in the year 1 800, and in
what months they happened.
3. Required the number of Eclipses in the year 1820, and the
months in which they happen.
To find on what fiart of the Globe any given Eclipse of the
Sun or Moon is visible.
This is most readily ascertained by means of an artificial
globe ; as follows :
The day and hour being given when a Solar Eclipse will hap-
pen y to find where it will be visible.
Precept. Find the Sun's declination, and elevate the pole
agreeably to that declination ; bring the place, at which the
hour is given, to that part of the brass meridian which is num-
bered from the equator towards the pole*, and set the index of
the hour circle to twelve ; then if the given time be before
noon, turn the globe westward till the index has passed over as
many hours as the given time wants of noon ; if the time be past
noon, turn the globe eastward as many hours as it is past noon,
and exactly under the degree of the Sun's declination on the
brass meridian you will find the place on the globe where the
Sun will be vertically eclipsed : at all places within 70 degrees
of this place, the eclipse may* be visible, especially if it be a
total eclipse.
Ex. On the 11th of February 1304, at 27 min. past 10 o'clock
in the morning at London, there was an eclipse of the
sun ; where was it visible, supposing the moon's penumbral
shadow to extend northward 70 degrees from the place where
the sun was vertically eclipsed ?
Ans. Britain, Ireland, France, Germany, &c;
The day and hour being given when a Lunar Eclipse will hap-
pen tofihd where it will be visible.
Precept, Find the Sun's, declination for the given day, and,
note whether it be north or south ; if it be north, elevate the
south pole so many degrees above the horizon a* are equal to
•Note. When the Moon is exactly in the node, and when the
atfis of the Moon's shadow and penumbra pass through the cen-
tre of the earth, the breadth of the earth's surface under the pe-
numbral shadow is 70° 20* ; but the breadth of this shadow is
variable ; and if it be not accurately determined by calculation^
it is impossible to tell by the globe to what extent an eclipse of
the sun will be visible.
ARITHMETICS. 2*fr
*lre declination ; if it be south, elevate the north pole in a simi-
lar manner ; bring the place at which the hour is given to that
part of the brass meridian which is numbered from the equator
towards the poles, and set the index of the hour circle to
twelve ; then, if the given time be before noon, turn the globe
westward as many hours as it wants of noon ; if after noon, turn
the globe eastward as many hours as it is past noon ; the place
exactly under the degree of the Sun's declination will be th6
antipodes of the place where the Moon is virtically eclipsed*
Set the index of the hour circle again to twelve, and turn the
globe on its axis till the index has passed over twelve hours ;
then to all places above the horizon the eclipse will be visible;
to those places along the western edge of the horizon the moon
will rise eclipsed ; to those along the eastern edge she will set
eclipsed ; and to that place immediately under the Sun's decli-
nation the Moon will be virtically eclipsed.
Example. On the 26th of January 1804, at 58 tnin. past seven
in the afternoon, at London, there was an eclipse of the Moon ;
where was it visible ?
dnswer. It was visible to the whole of Europe, Africa, and
the continent of A sia.
27
»
\
A
NEW AND CORRECT
TABLE OF LOGARITHMS
OF THE NATURAL NUMBERS FROM 1 TO 10.00#,
EXTENDED TO SEVEN PLACES BESIDES THE INDEX ; AND S*
CONTRIVED, THAT THE LOGARITHM MAY BE EASILY
FOUND TO ANY NUMBER BETWEEN 1 AND
10,000,000.
ARITHMETICK
2*7
>* -
1 U.JJJ'JJOO
2O.3JU30u
3047712i3
40.602J60J
5 0.6)8), oj
o0.//6i5io
70.3450980
8 0.9030906
9 0.954242o
10 l.OOv/OOOO
lM.im^;-
12 1.0791812
131 1139434
14 1 1461230
15 1 1760913
16 1-204.1JJJ
17 12304439
Id 1.2552725
19 1 2787536
20 1.301030J
No.
Log
N T O.I Loir i|X,
Lop-.
1.7075702 |
1.7160033
- 1.7242759
54)1.7323938
55; 1.7403627!
jo 1 74«18tt0 |
571.7558749'
5hjl 7634280 '
59,1.7708520 l
6017781513!
01,1.7853298:
62 1.8923917!
631.7993405 !
64 1.8051800 I
J35 1.8 12913 4 '
O6l.81954o3
67U.826074S
631.8325089
69(1.8388491
70 1.8450^)80 i
J i 1 322^i.'J
22 1.3424227
23 1.3617278'
241.3802112
25 1.39/Q409
< i 1.8512jo3
72 ( 18573325
73; 1.8633229
74 1.8692317
751.8750613
2o 1.4i4'J; JJ
2714313638
28 1447*580 :
29 14623980;
301-4771213;
31 149-ljol/.
32 1.5051500
33 1 5185139
34 1 5314789
.3) 1.54406H0
(o l.edJ8l6(>
77jl 8864907
78' 1.8920946
79,1.8976271
80.1.9030900
8i:1.90«4bi0 I
921.9138139
831 9190781 J
84^19242793 ,
85 1.9294189 !
oj \.5563\Jzj\
37.1.5682017!
38 1.5797835!
39 1.5910646"
40 1.60206.)0;
80 1.9344J8J
8719395193.'
881.9444827'!
89)19493900;
901954? t^v
+ • 1.61 j/ oj
42 1 62321.93
43:1.6.134685
44.1.6434527
45 1.6.5331 ?5
*->, I 6.32,- ^c .)
47! 1.5720979
4816812412
49.16901951
5oil 696') 7 00
yi 1.95r/j-i: •<• ,
921.963:378 j
93 196848291
941. 73127?
95l9/7'2-
97 1986771",
981.9912261
99 j 1995 53 52! I
1U.r20j0iOK)|
01 2.0043214
02 2.0086002
.03^2.0128372
104;2.0170333
05'2.0211893
oo 2.0253059
0720293838
08 2 0334238
09 2 0374265
10 2 0413927
11 2.045oJ50
12 2.0492180
13 2 0530724
14 2 0569048
15 2 0606978
lo 2.0o44j6J
17 2 0681859
18 2 0718320
19 2 0755470
20 2 0791812
zl 2 0827654
22 2 0363598
23,2.0899051
24 2.0934217
25 2 0969100
26;2.1003/U5
27,2.1038037
28,21072100
2912 1105897
30,21139434
31,3 1172/13
32|2 1205739
33;2123£516
34:21271048
35|2 1303338
o6;2 1335389
3721307206
38| 21398791
.3912.1430148
40i21461280
4l|il492i^i|
42215228831
43 I 21553360|
44J215S3625
45|2.16136rt0i
46: 2.1 64352 w
47 21 673173
48121702617
4i:| :.i73iS3o
50|2 176;)913
51:21789769
52J21818436
53J21846914
5412.1875207
55i21903317
56:2 19312*0
5721958997
58 21986571
59'2 2013971
602.2041200
Ln R .
201 '2.3031961
202 2.3053514
203 2 3074960
204 2.3096302
205 2.3117539
206 2.3138672
2072.3159703
208 2 3180633
209 2 3201463
210 2 3222193
61 2.2w6o2oy
6212 2095150 '
63 2 2121876
6412 2148438
65:2.217483'
06:2.2201081
67;2.2227165
68;22253093
69,22278867
702.2304481-
71 2.2329961
72 2.2355284
73.2 2380461
742 2405492
7512.2430380
2ii2o242tt25
212 2.3263359
213 2,3283796
214 2.3304138
215 23324385
2i6 2o344j38
217 2 3364597
218 2 3384565
219 2.3404441
220 2.3424227
221 2 3443923
222 2 3463530
223 2.3483049
224 2.3502480
225 23521825
762 245512,
7712.2479733 .
78 2 2504200 j!
792 2528530 <
80 2 2552725 ['
81 22576786
82 2.2600714
83 2 2624511
84 2.2648178
85 2.2671717
226 2.3541084
227 2.3560259
228 2.3579348
229 2.3598355
2 iO 2.3617 278
86 2.2695129
872.2018416
88 2.2741578
89 2.2764618
90 2 278753 6
yl 22810334
92 2.28*301
93,2 2855573
94 2 287801
95 229.00346
yoj2 29226o7
97!2.294466?
98 2.2966652
99 2 2988531
2002.301030(
2ol ,2 3636120
232 2.36,54880
233'2 3673559
234J2 3692159
23512.3710679
"236-2 37 29120
237J2 5747480
2382.3765770
2392.3783979
24023802112
241 2.3.820170
242 2.38o8154
243 2-335 06.:
244 2 3873898
245 2 38M661
246 J.59u9o6l
247 2.3926969
248 2.394451
249 2.3961993
, 250 2 3979400
I
80S
LOGARITHMICK
"No,
25l
2522.
253
Log.
2.3996737
.4014005
2.4031205
259
260
261
263
264
2542.404833;
255 24065402
25624082400
2572.4099331
25824116197
2.4132998
3.4149*33
2.4166405
2622 4183013
2.4199557
2-4216039
265 ! 2.4232459
366j2.4248816
267
2.4265113
269
270
2682.4281348
2.429752,
2.4313638
2/1:2.4329690,
272 2.4345689!
273|2.4361626
27412.4377506
275 ! 2.4393327
27624400091
277|2.4424798
27824440448
279.2.4456042
280; 2.4471580
2^4487063
24502491
2.4517864
2842.4533183
2852.4548449
281
282
283
286
287
288
2.4563660
2.4578819
24593925
289'24608978
2902 4623980
291
292
293
294
295
2.4638930
2.465382?
2.4668676
2.4683473
2.4698220
2962.4712917
29724727564
298 2 4742163'
299 24756712
3002.4771S13
"301
302 2.
304 2.
303
304
305
306
307
3082.
309
310
Log.
2.4785665
1.4800069
2.4814426
.4828736
2.4842998
2.4857214
2.4871384
.4885507
2 4899585
2.4913617
311
312
31
314
315
2.4927604'
2.4941546
24955443
2.4969296
24983106!
316
317
318
2.4996871|
2.5010593 :
„ 2 50242711
3192.5037907
320!2.5051500»
32112.50650501
322'2.5078559!
323;2 5092025|
32412.5105450
325(3.5118834!
32612.5132176
3272.5145478;
328 2.5158738!
329 2.5171959;
330:2.5185139!
331
332
333
2.5198280
2.5211381
25224442'
33412.5237465
33512.5250448
336
337
338
2.5263393!
?. 5276299
2.52891 o7
339 2 5301997
340 2.5314789
341
342
343
344
345
2.5327544
2 5340261
2.5352941
2.5365584
2.5378191
346
347
348
349
350
2.5390761
2.5403295
2.5415792
2.5428254
2.5440680
No. Loy:.
3512.5453071
35225465427,
353 2.5477747
35412.5490033
355 2 5502284
356
357
358
359
2.5514500
2.5526682
2 5538830
2.5550944
3602.5563025
361
362
363
364
365
366 2
367
368
369
370
371
372
373
37412.
375
2.5575072
2.5587086
2.5599066
2.5611014
2.5622929
5634811
2.5646661
2.5658478 (
2.5670264
2.5682017
2.5693739
2.5705429
2.5717088
572871^,
2.5740313
3762.5751878
3772.5763414
378
379
380
381
382
383
3842
385
386
387
388
389
390
391
392
393
394
395
39/
398
399
400
2.5774918
2 5786392
2 5797436
2.5809250!
2.5820634!
2.5831988
5843312
2.5854607
2.5865873
2 5877110
25888317
2.5899496
2.5910646
2.5921768:
2.5932861,
2.5943926
25954962
2.5965971
396(2 5976952
2 5987905
2 5998831
2.6009729
2.6020600,
No. | Log.
401 2.6031444
4022.6042261
4032.6053050
4042.6063814
405J2 6074550
4062.6085260
407J2 6095944
4082,6106602
409J2.6117233
4102.6127839
4112.6138418
41212.6148972
413 2 6159501
4142.6170003
415'2.6180481
4162.6190933
417|2.6201361
4182.6211763
4192.6222140
4202.6232493
421 2.6242821
422
423
424
425
426
427
428
429
430
431
432
433
434
435
4362
437
438
439
440
441
442
443
444
445
446
447 2.
449
450
2.6253125
2.6263404
2.6273659
2.6283889
2 6294096
2.6304279
2.6314438
2.6324573
26334685
2.6344773
2.6354837
2.6364879
2.6374897
26384893
;!
No.f Log.
6394865 1
2.6404814;
2.6414741i
2 6424645'
26434527!
26444386
2 6454223
26464037i
2.6* 73830!
2 6483600
4512.6541765
4522.6551384
45326560982
454 2.6570559.
455*2.6580114
4562.6589648-
457J2 6599162
458 2.6608655
459.2.6618127
4602.6627578
4612.6637UU9
4622.6646420
46312.6655810
464J26665180
465 ! 2.6674530
.'66|2.666ob59
46712.6693169
468*2 6702459
469J2.6711728
47012.6720979
471
472
473
474
475
2.6730209
26739420.
2.6748611
2.6757783
26766936
476
477
478
479
480
26776070
2.6785184
26794279
2.6803355
2.6812412
481
482
483
484
485
487
488
489
490
2.6821451
2 6830470
2.6839471
2.6848454
2 6857417
486 2.6866303
26875290
2.6884198
2.6893089
2.6901961
491,26910815
4922.6919651
4932.6928469
4942.6937269
495,2.6946052
2.6493349
.6503073
448 2.6512780
2.6522463
2.6532125!
496 2-6954817
497
498
499
500
2.6963564
2 697229:
2.6981005
2 69897Q0
ARITHMETICS
409
501
502
503
504
JJ05
50b
507
508
509
510
611
512
513
514
515
"51b
517
Lotr.
2.699837!
2.700703
2 7015680|
2.7024305
270329H|
2.7041505!
2.7050080
27058637
2.7067178!
2 7075702
27084209
2.7092700
2 7101174
2.7109631
2.7118072;
2.7126497!
2.7134905
51827143298^
519J2.7151674 1
^201271600331
521J27168377I
5222.7176705
52312.71850171
524i27193313
525; 2.72015 93-
526|27209857i
5272.7218106
5282.7226339
529;2 7234557S
5302.7242759
No. Log. No. Log.
"55T2.74ll5l6f"B0l2!
552 2.7419391 602 2.
553 27427251 603 2
55427435098 604
555 27442930 605
556 27450748, b06
557 27458552' 607
5582 7466342! 608 *■
559 2 7474118! 609 2
560 2.7481880; 610 2 -
561 2.7489629' 6li- 2
562 27497363J, 619 2
563 27505084|! 613 2
564 27512791!| 614 2,
565 27520484 ' 615 2.
566 2 7528164;i~bl^2
567 2 7535831' 617 2
56827543483 618 2
53127550945;
532'2 7259116
53327267272
534 27275413
53527283538
~536 272yi64d
53727299743'
538 27307823
53927315888
540 27323938
541 27331973
542 27339993'
543 27347998 1
544 27355989i
^45 27363965'
546 2.7371926!
547.27379873
543 27387806
549,27395723
55027403627
7788745
!7795965
7803173
7810369
7817554
7824726
7831887
7839036
7846173
No
651
652
I 65,
654
655
006
65,"
658
650
;'2.81358:
,|No. Lo<-\
8135810 701
2.8142476
2.8149132
2.8155777
2.8162413
2 8169038
.'.8175654
2.b782259
2.8188854
569 27551123
570 27558749 ,
571 275663bl
57227573960.
573 27581546
574 27589119
575 2.7596678
576 27604225,
577 27611758!
57827619278
579 27626786
580 27634280
619 2,
620 2
T5l£
622 2
623 2.
624 2.
625 2.
631(2
632 2.
633 2.
634 2,
635 2
~S36 2~
637 2.i
638 2
639 2.
581 a./tku/oi-
5822.7649230
58327656686
584J27664128
_585!27671559
585i27b78y7b
587'27686381
588J2 7693773
589J27701153 w *
. 590;277085f>0 640
^i!27715b75
592,2.7723217
593I2.7730547
594:27737864
_595i 2 7745170
596, j ., ( j^^oo
597J27759743
598J27767012
599127774268
600127781513
785329^ 660 J 2 HI' '543?
78604121 66i
7867514! 662
78747051 663
7881684!: 664
7888751!' 66p
78958U/, 66b
667
668
669
67C
2.8208580
2.8215135
2.8221681
2.8228216
7902852*
7909885;
7916906!
79239171
"626 2.;
627 2.
6282.
629 2
630 2
7930916
7937904
7944880
7951846
7958800
0412
642 2.
643 2
644 2
645 2.
646 2.
647
648 2
645 2.
650
7y6574o
7972675
7979596
7986506
7993406
J.8007171
J8014037
!.802089
18027737
8034571
8041394
2 82ot<-*ii
2.&241258
2.8247765
2 8254261
2.8260748
b71 2 8267225
672 2 8273693
673 2 8280151
674
2.8286599
675 2 82930^9
676
677
2.8305887
682
683
687
8048207J1 688
80550091 i 689
2.8061800J1 690 2.8388491
.8068580! 691
.80753501; 692
678 2.8312297
679 2 8318698
680 2 8325089
"681
2.&5ol4/l
2.8337844
2.8344207
6842.8350561
685 2.83569Q6
686 2.836o241
2.8369567
2 8375884
2.8382192
69.
694
695
8082110;
.8088859J
■8095597 | __
.81U2o25!J~6'?C
.8109043! 697
.8115750, 698
812244711 699
.8129134' 1 700'
2 8394780
2 8401061
2.8407332
2.841:3595
.8419848
2b4^0U92
2.3432328
28438554
2.8444772
2 8450980
703
704
705
2.84o/lii(j
702 28463371
2 8469553
2.8475727
2 8481891
t wo 2.8488047
70712-8494194
708 , 2.850033S
70^!2.85i>6462
71u'2 8512583
712 2 8524800
71312.8530895
714J2.8536982
M 5 2.8543060
717
718
719
720
721
722
723
724
725
.*.<x/4i*loo
28555192
28561244
2.8567289
2.8573325
2.85*79353
2 8585372
2,8591383
28597386
?.86«»3::80
^.oouybbb
2 8615344
2.8621314
2 8627275
2.8633229
726
727
728
729
730
~73lJ2.8639174
732I2.8645111
733|2.8651040
734i2.8656961
735 j 2.8662873
736^.tt008778
7372.8674675
738'28680564
739J28686444
74028692317
741;2.8698182
74212 8704039
743J2.8709888
74428715729
745J28721563
748
749
750
2.8739016
2 8744819
2.87506131
21*
LOGARITHMICK
•Mo. Li-.
751
7j\j
753
75
,N«
Lo;>
f!
2 87621/8
8.V3/1-3
755 2 8779470 ;
rju[2.orB5jiv5 1
75/U-&I9oJ5J\
75$28r965J2
7a9J2.ittO.j4 18
7tf>'2ck>J.Si ,\>
/ v>i j2 OJijOt,'
762 2.86U 550
763; 2 88 252 ±.5
7iH>2 8330i>3 J.
/6V2.8336614
76712.8847954
7 6 si 2.6X536 12
769'2 6559263
7702886*907
771 2.88/0544
772J288/6173
773\2 8831795
77*2.838/410
775 2 6693017
776j2^898ol/
777 2.8:W4210
778;28909796
779f2 89153/5
78^2- 8920946
78iJ2:8J26510- ;
782 [2 8932058
783J2893761S
784J2 8943161
765 '2 8948607
787j2 895J747
766 2 6955262
789J2 6970770
790i2 89 76rl
8Jii2:90o , jj^\\
802|2.9u4l744j
803 2.9047151 I
804 ! 2 905256o' !
803 2 905/959
80^2.90633^0 '
8 J7 29068735'
808 2.9074114^
800 2.9079485
810 2 908485U
8112.90002^;;'
812 2.9595560:
813 2.91009U5 i
8U 2.9106'JH !
8152 91115/ 61
8V3 2.9llo''j}\
til/' 2.9122221 1
8 18'2 9127533!
819 2.yi328.i0j
820 2 9135139
821.2.9143432
822 2.9148718
823 2.9153998
824 2 9159272
825 2 916*53?
8512.9299296
852 2.9304396
853 2.9309490
8542.9314579
855 2 9319661
6j j 29324738
857 2 9329808
858 2 9334873
85929339932
860 2.9344985
.so.
901
902
903
904
905
906
Lo^.
2.9547248
29552065
2.9556878
2.9161684
2.9566486
'X9571282
826 2 91 u Jo J o
827.2.9175055
8282 9180303
829 2 9185545
830 2.9190781
c^i 2.9196010
832 2 9201233
833 2 9206450
8312 9211561
83? 2.921 •W-iri
8ol 2.9350032
8622.9355073
863 2.9360108
864 2.9355137
865 2.9370161
866 2.9375179
l 8? 2-9380191
868 2.9385197
869 2.9390198
.8702.9395193
& n '2^9400182
8 ' r 2,2.9405165
8732.9410142
874i2.9415114
jj£jJ2.9420081,
8/6 2^9425041
877J2.9429996
8782.9434945
879 2.9439889
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881 2948975,'
882 2.9454685 1
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884 2.94S4523I
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7"J5. t 2. 0036M
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7^8(2.9020020
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837 2.92272 5.3
833 2.9232410
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842 2.9253121
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890 2.9493900
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9ni2.959.il84
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9132.9604708
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913 '2 961421 1
916,2,96^8955
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918.29618427
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2 9637878
2.9642596
2 9647309
2.9672017
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2 96661 10
2 9670797
2.9675480
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2.9698816
2.9703469
2.970*116
891 2.94987/?
892 2 9503649
893 2 9508515
894 2.9513375
895 2.9518230
896 2.95i Ju8U
897 2.952792
898 2 9532763
899 2.953750'
900 2.954242:
920
921
922
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92j
92,
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928
929
930
931
932
933
934
935
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937
938
939
940
941
942
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947
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952(2.9786369
953,2.9790629
95429795434
9552.9800034
956 2.9804579
957 2.9809119
958 2.9813655
959 2 9818186
96Q'2.9822ag
961 2 9827234
922 2 9831751
963,2 9836263
9642.9840770
965, 2.984527 :
966 : 2.9849771
967;29854265
968 2.9858754
969:2.9863238
970 2 9867717
■97i'2.99'72192
9722.9876663
973 ! 2 9881123
974 2...835590
9752.9890046
976J2.9894498
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97812.9903389
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980,2 9912261
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982J2.9921115
98329925535
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2.9<iZ(o6
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2.9722028
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990 2.99563 :
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992 2.9965117
993 29969492
994 2.9973864
995 2.9978231
990 2.9982593
9972.9986952
998 2.9991305
999 2.9995655
1000 3 0000000
ARITHMETICK.
211
iou5'o.Oo259bU 1056;o.
1007130030295! 1057(3.
lOOtS 3 0034605 1058'3.
1009I3-0038912. 1 059*3
1009,^038912 10593
1010 3 0U43214 iio6o3.
lull 3.0047512 lObl'S.
1012 3 0051805 1062 3-
1013 30056094;i063 3
101430060380 10643
1015 30064660 J1Q65 3
•l066 3
K^6 3.0068937
1017:3.0073210
10183.0077478
101913.008174:
1020 3 0086002
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10053 0021661?ji055 i 3
Nf'.j Lo<j.
0216027
0220157
0224284
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0232527
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0253059
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1101
1102
1103
3.041787*:
3.0421816
3.0425755
1104.3.0429691
1105!3 0433623
1106:
1107!
1108
1109
1110
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3.0441476
1044539b
3.0449315
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0285713 1118
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3 0457141
5.0461048
3.0464952
3.0468852
3.0472749
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3.0476642
30480532
3.0484418
3.0488301
3.0492180
10263.011 1474,1076 3
1027 3.01 15704' jl077'3
1028'3 0119931s!l0783
10293 0124154' 1079 3
103o3.Q128372l080 3
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.03H085 1125*3 0511525
0318123 1126 3 0515334
,0322157 1127;3.0519239
03261881128
0330214 1129
0334238 1130
S.0523091
3.0526939
0530784
10313.01325871081
1032 3 0136797j 1082
1033,3.0141003j|1083
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30338257
3.03:2273
3 0346285
3.035029:
1035 3.0H94O3 : !lO85 3 035421V
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1155 3.f;fr7.ifc2o 1205 : 3 0809870
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1042 3017867-
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3.0576661
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3.0692980
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3.0700379!
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1222 3.08; 07 12
12233.0874265
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{
212
LOGARITHMICK
No. | Lo<. Mu.| Lo£.
125130972575
1252 3.0976043;
1253;30979511|
1254*3.0982975
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1017471
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1351
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1303 3.114 444 13533.1312978 1403
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1309 3.1169396
13103.1172713
131631192559
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13223.1212315
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1325*3.1222155
132631225439
13273.1228709
13283.1231981
1329
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3 1238516
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1357»3.1325798j 1407 3.1482941
13583.1328998J 1408 3 1486027
12593.1332195 1409 3.1489110
1360J3 13353891 1410 3.1492191
1361|3.1338581i 1411
13623.13417711 1412
1363'3.1344959J1413
1364J3.1348144| 1414
1365 ! 3.1351327j|1415
3.149527.0
3.1498347
31501422
3.1504494
1366 3.1354507i 141613
1367|3.135768ff7l417
1368^1360861'! 1418
13693.1364034 l jl419
137031367206;>1420
1371|3.1370375
1372 3.1373541
13733.1376705
1374 ! 31379867
1375:3.1383027
13763.13861841
1377i3.1389339
137831392492
1379J3.1395643
1380 3.1398791
1331 3.124178iri38l3.1401937
133231245042 [1382 3 1405080
1333 31248301 1 13833.1408222
1334 3.1251558 .1384;31413612
13353.1254813 13853.1414498
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1392:3,
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13953.
1433271
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1439511
1442628
1445742
1421
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1423
1424
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1426
1427
1428
1429
1431
Lofr.
No.) Lojtr.
1451 3.1616674
14523.1619666
1453 3J622656
1454,31625644
1455 3.162863
14J631631614
1457 3.1634595
1458:31657575.
1459|31640553
1460|31643529
14613.1646502
1462*3.1649474
1463J3.1652443
14643.1655411
3.1507564' 1465:3 1658376
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31513699!;1467 31664301
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3.1532049
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31541195^
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142831547282
3.1550322
143031553360
3.1556396
14323.1559430
143331562462
1434:3.1565492
1435 , 3.1568519
1436,31571544
143713 1574568
143831577589
143931580608
144031583625,
1441 3.1586640s
1442 ! 31589653
1443 31592663
1444 31595672!
1445 3.1598678;
134631290451 13963.1448854
1347(3.1293676 1397,3.1451964
1348'31296899 1398 3.1455072
13493. 13001 19 11399 3.145H177
13503 1303338 11400 3.1461280
31676127
147231679078
1473 31682027
14743.1684975
147531687920
1383
1484
1485
31670218
147631690864
1477 31693805
147831696744
1479 31699682
148031702617
1481 3.170555X
1482 3170848S
3171141:
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3.1717265
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1487
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172311(3
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144731604685 1497J3 1752218
1448316076861 149831755118
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14503 1613680'|1500l3.1760913f
ARITHMETIC*,
SIS
Nfo.| Log^ ! No.
I501|3.176j«u/ i 1551
1502 3.1766699 1552
1503 3.1769590 1553
1504&1772478J1554
1505 3.17753651(1 555
3.1804126
15163.1806992
151731809856
1518
15193.1815578
15203.1818436
1506 3.1// 8i>oU
15073.178113S
15083.17W01
15093.1786892
1510*3.1789769
15113.1792645
1512
1513
3.1795518
3 1798389
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1522
1523
15143.1801259
1515
3.18212*2
3.1824147
3.1826999
11556
!l557
1558
1559
! 1560
1561
1562
1563
„og.
3.1906118
3.1908917 (
3.1911715
.1914510
1917304
3.1920096
1922fc86
31925675
3.1928461*
3.1931246
3.1934U29
31936810
3.1939590
15643.1942367
3.1945143
1565
iswi'o.iiMfmtt
1567|3.1950690
3.1812718| 1568 3.1953461
15693.1956229
15703.1958997'
1606
1607
1608
1309
1610
1611
1612
161
1614
1615
1616
1617
1618
1619
1620
57113.1961762,
'1572 ! 3.1964525
1573|3.1967287|
1574 , 3.1970047
1575J3.1972806
1625
32108534
157613.1975562
157713.1978317;
15783.1981070
15793.1983821
1580 3.1986571
lb*
1627
3.2119211
3.2121876'
15813.1989319,
1582 3.1992065!
!1583!3.1994809'
158431997552,
15JB5| 3.200029 3
15863.2003032
1587;3.2005769
153835008505
!l589!3.2011239j
15903.2013971 1
3.2111205
12113876
1628|3.2H6544
1629
1630
1631
1632
1633
1634
1635
3.2124540
3,2127202
3.2129862
3.2132521
3 2135178
I5f6 3513783;
1 15243.182985Q
15253.1832698
1526:3.1835545
152731838390
15283.1841234
15293.1844075
1530 31846914
S53I 3.1aW:K
15323.1852588
1533'3.1855422
153413.1858254
1535 3.1861084
I53S ! 3.1863912
15373.1866739
15383.1869563
393.1872386
15403.1875207
1541 3.1878026 159l|3 2016702| T641
15423.1880844 15923.2019431' 1642
1543 3.1883659 11593 3.20221581 j 1643
1544 3.1886473l!l594 3.2024883;! 1644
1545 : 3.1889285 iil5953 : 2027607lll645
1546 3.1892095,110^6 3.2030329,1046 >5l642!*8 j
1547 3.1894903 1 1597,32033049: 1647 3 2166936 I
1548 3 1897710 1598 3.2035768 1 ; 1648 3.2169572||
1549 3 1900514 159935038485', 1649 3 5172207|'
15 50'319Q3317'll6QO 3.204 1200 1650 3.2174839,
28
1605
Lop.
No.
lS6I 3.2043913!
16023.2046625
1603 3.2049335
16043.2052044 1
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o.^Uj/4d5,
3.2060159!
2062860'
3.2065560
2068259,
o.2U/Ui*jj
3.2073650
3.2076344
3.2079035
3.2081725,
.2084414
3 2087100
3 2089785
3.2092468
3.2095150
1621 3.2097830
1622 3.2100508
1623 35103185
1624 3.2105860
1637
1638
1639
1640
3.2053044
3.2062860
3.2065560
3 2068259!
o.2 la 1086
35153732'
35156376;
32159018
35161659 1
No./ Log. ijNo,
rW135T7747ii
6523.2180100
3.2182729
653
6543.2185355
655
32187980
66;
664
665
666
667
668
669
670
1701
1702
1703
3.230704^
3 2309596
1.2514146
1705
3563.21906031 l'M>
657
658
659
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661
17043.2314606
3.2317244
S^owyo
3.2193225 Il707 3.2322335
3 2195845 J1708 3.2324879
3.2198464 1709
35201081 171?
3.220369611711
6623.2206310 !171i
35208922 ;1713
35211533 11714
35214142,1715
3.2216750! 1716
3 2219356*1717
35221960
3.2224563
35227165
oil
672
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35229764
223236;
35234959
675
67435237555 1
35240148
o/o«o 2242740
677
678
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3.2247920
67935250507
35253093
680
681 35255677
68235258260
35260841
2263421
35265999
689
690
b'jl
692
69;
694
695
696
697
699
1731
1732
1733
1734
1735
3.2268576l?ob
1718
1719
1720
1721
683
684
£85
686
68735271151ill737
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3.2521246
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9133.2817150
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927
3.2846563
3.2848817
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3.2853322
3 2855573
1932
933
934
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9313 2857823
3.2860071
3.2862319
3.2864565
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1949 ! 3.2898118
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No.| Log.
19513.290257
195232904798
1953 3.2907022
19543.2909240
19_5_5'3.2911468>
1956I3.2913689
1957J3 2915908
1958 3.2918127
1959 3.2920344
1960 3 2922561
1961
1962
1963
4966
1967
1968
1969
1970
3.2924776
3 2916990
3.2929203
1964,3.2931415
19653 2933626
3.2935835
3.2938044
3.2940251
3.2942457
3.2944662
1971 3.2946866
1.972J3 2949069
1973 3.2951271
1974 ! 3 2953471
197513 2955671
19763.2957869
1977
1978
3.2960067
3.2962263
19793 2964458
1980 3 2966652
laol',3 2968845
1982'3.2971037
1983 3.2973227
1984'3.2975417*
1985 3.2977605
1986
1987
1988
1991
1992
199:
0.2*79792
3.2981979
3.2984164
19893 2986348
19903.2988531
3.24/90713
3.299289:
3.2995073
1994,3.2997252
1995 3 2999429
199b
1997
1998
1999
2000
O.5UU1605
33003781
3 3005955
3.30081 2b
3.301030C
ARITHMETICK
«15»
>».,
Log.
2001 3.3012471
20023.3014641
2003 3.3016809
20043.3018977
20053.3021144
2006 3.3023309
20073.3025474
20083.3027637
2009 3.3029799
20103 3031961
20113.3034121
2012 3 3036280
2013 3 3038438
20143 3040595
2015 3 3042751
2016 3.3044905
10173.304705^
20183.3049212
20193.3051363
2020 3.3053514
2021 3.3055663
20*3.3057812
2023 3 3059959
20243.3062105
202533064250
2026 33066394
2027 3.3068 37
2028 3.3070680
20293.3072820
20303 3074960
No.
2051
2052
LOVT.
3.290257.
2904798
2053 3 2907022
21U1
2102
2103
2104
2105
2056(3 2913689 |2106
2054
3.2909240
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3.3224261
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3.3228393
3.3230457
3 3232521
3.2915908 2107
2918127,12108
3.2920344 2109
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No. I Log.
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22043 3432116
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2206 0.343 6055
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2228 3.3479152
2229 3.3481191
2230 3.3483049
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3782161
3783979
3785796
3787612
3789427
3791241
3793055
3794868
3796680
3798492 1
3800302 1
3802112
No. I Loj*
24063
2407;
2408
24093
2410
3803922!
3805730'
3807538
3809345
3811151
3812950,
3814761
3816565
3818368
3820170
24113
24123
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2416.3
24173
241S3
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24203
38219/2
3823773
3825573
3827373
38 29171
3830969
3832766
3834563
3836359
3838154
242113
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24233
24243
24253
3839948
3841741
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3845326
3847117
2426'3
24273
2428,3
2429;3
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2431j3
2432*3
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24403
3848908
3850698
3852487
3854275
3856063
3857850
3859636
3861421
3863206
38 64990
3866773
3868555
3870337
3872118
3873898
24413
24423
24433
24443
24453
3S75678
3877457
3879235
3881012
3882789
No.
Lo&.
24513
2452
24533
24543
24$5 3
3893433
3895205?
389G975
3898746
3900515
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24573
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3902284
3904052
3905819
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24693
24703
2471
24723
2473
24743
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24763
24773
24783
2479
2480
3911116
3912880
3914644
3916407
39 18169
3919931
3921691
3923452
3925211
3926970
3928727
3930485
3932241
3933997
39357*2
3937506
3939269
3941013
3942765
3944517
2481 S
24823
24833
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1 2485
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2447,3
24483
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2450 3
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3886340
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3891661
24863
24873
24883
24893
24903
24913
2492
2493 3
24943
2495 3
2496
2497
2498 3
2499
2500 1
3946268
3948018
3949767
3951516
3953264
3955011
3956758
3958504
3960249 t
3961993
3963737
3965480
3967223
3968964
3970705
3972446
3974185
3975924
3977663
3979400
ARtTHMETICK.
tit
2503
3504
25053.3988077
25063
2507
250813.
2509
2510
2511;3.y998467
2512;3.4000196
2513'3.4001960
25143.4003652
25153.4005380
2516 ! 3.4007106 J2566
2517-3-4008832 '2567
2518-3 4010557 ,2568
2519 3.4012282* 2569
2520:3.4014005, 2570
2121J3 4015728
35223.40174511
2523:3.4019173 2573
252434020893 2574
No
2501
25023.3982873
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33981137
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3.3986343
12554 3.
2555
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33991543
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3.3995007
3.3996737
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No. Lop:. I No.l LoffV
2551
3.4
.4067105
2552)3.4068807,
,4070508
,4072209
.4073909;
2553
3.<
12556
2557
2558
12559
2560
2561
:2962
2563
2564
3.4084096
3.4087486
.4089180!
2565 3.4090874
2571
2572
2573
2525;34022614
2575
25263.4024333
2527)3.4026052
252834027771
2529|3,4029488
25303.4031205
2531-3.4032921
2532:3.4034637
2533!3 4036352
2534 ! 3.4038066
2535i3.4039780
25363.4041492
!537|3.4043205
t538!3.4044916
253913.4046627
2540 ! 3 4048337
2541
2542
2543
25443.
8545
3 4050047
34051755
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.4055171
34056878]
2546 3 4058584
2547;34060289
25483 4061994
2549)3 4063698
2550 ! 3 4065402
2576
2577
2581
2582
2583
25843.
J25863.
J2587 "
2589
2590
2591
,2595
2590
2597
2599
2600
.4075608
.40773071
.4079005
.4080703-
.4082400;
26013,4151404
26023.4153073
26033.4154742
26043.4156410
2605(3.4158077
26ub
2607
2608
2609
2610
2611
3.4168009
.4085791,126123.4169732
34171394
3.4173056
3.4174717
34092567
34094259
3.4095950
3.4097641
3*4099331
34101021
34102710
3.4104398
3-4106085
S 4107772
3 4109459
3.4111144
25783.411.?829
3.4114513
25803.4116197
3.4117880;
25823.41195621
3.41212441
.4122925J
25853.4124605,
,4126285
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258834129643'
3.4131320,
3.4132998 1
34134674
25923.4136350
21933.4138025
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3.4141374f
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25983.4146391'
3.4148063
3.4149733
34159744
3 4163076 !
3.4164741
34166405
2613
2614
2615
2616
2617
2618
2619
2620
1265634242281
4161410 2657.3 4243916 1
2658342455501
12659 3.4247183
!2660 3.4248816
5.4176377
3.4178037
34179696
3.4181355
3.4183013
3.4184670
3.4186327
3.4187983
3.4189638
2621
2622
262
2624
2625|3.4191293
2626&.4192947
2627J3.4194601
262834196254
262934197906
26303.4199557
263113.4201208
2632(3.4202859
2633i3.4204509
253413.4206158
26353.4207806
2636134209454
2637|3 4211101
2638'3 4212748
2639J3 4214394
2640 ! 3.4216039
2641
2642
2643
3.4217684
3.4219328
4220972
26443 4222614
2645 3.4224257
<J040
2647
2648
2649
3.4225898
34227539
3.4229180
.4230820
2650 3.4232459
No.) Log, j
No. | Log- -
26513.4234097
265234235735
26533.4237372
26543.4239009
26553.4240645
270113.4315246
2702 3.431685$
2703i3.43l8460
27043.4520067
2705 3 4321673
27063.43232?8
£707 ! 3.4324883
27083.4326487
2709 ! 3.432809O
2710 ! 3.4329693
2661 3.4250449;
26623.4252031
26633.4253712 1
26643.4255342
2665,3.4256972 ;
26663.4254601!
266713.4260230
26683.4261858
2669 ! 3.4263486'
2670 J 3.4265113 ;
26713.4266739
2672:3.4268365
2673'3.4269990'
2674'3.4271614i
26753.4273238J
26763.4274861
26773.4276484
26783-4278106
2679!3.4279727
26803.4281348
^7113.4331295
2712'3.4332897
27133 4334498
2714|3.4336098
2715 3.4337698
27 16 3 .4339398
2717
2718
2719
2720
3.4340896
3.4342495 i
3.4344092
3.4345689
27213
2722
2723
2724
2725
,4347285]
3.4348881
3.4350476
34352071
3.4353665
2726i3.4355259
272713.4356851
2728 ! 3.4358444
J2729 3.436003*
2730134361626
268113.4282968
2682 3.4284588
2683 3.4286207
26843.4287825
26853.4299443
2731!3.43t>3217
2732 3.4364807
i2753 13.4366396
12734 3.4367985
2735 3.4369573
U7o6
2737
;2738
26893.429590812739
26863.4291060
2687J3.4292677
2688^4294293
2690*3 4297523
2691«3.4299137
26923 4300751
269334302363
26943.4303976
2740
3.4371161
33372748
3.4374334
.4375920
3.4377506
2741
2742
2743
26953 4305588 J2745
27443.4383841
269634307199
269713.4308809
2698'3 4310419
269934312029
27003 43136381
2748 i
2749
2750
3.4379090
34380674
34382258
3 4385423
(2746 J.438? 005
27473 4388587
3.4390167
3.4391747
13 4393327
• 18
LOGARITHMIC*
27513.4394*Uo|
27523.4396484!
2753 3.43980621
27543.4399639j<2304 , 3.
2755 3.44G1216|2305 ! 3
2/ou 3 44U2/*4 28UO 3
2757 3.4404368 2S07\3.
27583 4405943 2808 3.
27593.4407517 2809,3.
2/603 4409091 ;281 03
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27623.441223
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27643 4-115380
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2767 '3.4460092
2768 !3 4*21661
2769)3.4423230
2770!3.442479£
2771
27723.4427932
No.| L'j*.
No.| JLoi*. i,iNo. Lo£.
2*Ul|3.4473ioi 2851 3.4549972
2802'3.4474681« 2852134551495
2803'3
4476231' 2&5S
4477780 1 2854
4479329 2855
44808// 28563
4482424 2857
.4483971' 2858
,44355i7j 2859
4487063 2860
3.4426303
277:
2774
2775
3.4429499
34431065
3.4432630
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2780
2776 3.44341*5
27773.4435759
27783 4437322!
3.4438885
3.4440448;
281o3.
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28183.
28193.
2820 3,
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2828-3.
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2830.3
44660UO 2801
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4491697 2863
4493241 |2864
4494784 '2865
2860
12867
2868
2369
2870
3 4565179
3 4556696
3.4568213
3.4569730
3.4571246
4496327
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4499410
4500951
4502491
4504031
.4505570
.4507109
4508647,
4510185
3.4574/62
3.1,574277
3.4575791
34577305
3 4578819
4511722
3413258
4514794'
.4516329
.4517864
2876
2877
2878
2879
2880
2781 3:444201012831 3.4519399,
2782 3.4443571 2832 1 3 4520932
2783 3.4^45132!!2833;3 4522966
2784 3 4446692;!2834'3 4523998
2785I3.4448252I 2835 3.4525531i
2786;3.4449821i
2787|3.4451370
27883 4452928 1
2789.3.4454485J
2790*3 4456042!
27913!
2792J3
2793 j 3
27^4 ! 3.
2795^3
2836
2837
2838
2839
2840
.4457598;
4459154J
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2831
2882
2883
2884
2885
3.4527062
3.4528593 1
3 4530124'
3.4531654 2889
3 4533183
2841(3.4534712
28423.4536241
2843'3.4537769
234t|3.4539296
28453.4540823
2796344653^2 2846 3.4542349
2797J3.4466925 2847 3.4543875
27983.4468477i 28483.4545400
2799 3.4470029, 28493 4546924
58003.4471 580, 12850 3.454S449 1 .
2886
2887
2888
2890
2891
2892
2893
2894
2895
3.4553018
3.4554540
34556061
No.f Lo£. |,No|
455/582
3.4559102
3.4560622
3.4562142
3 4553660'
2901 3.46254/71,2951 3.4699692
2902 34626974 1 2952 O.4701164
2903 S.46^8470 1 2453 3.4702634
2904'3'4629966 '29543.47041Q5
!2905 34631461 'j2955 3A70557 Z
2906 , 3.463295b,-2956 3.4707044
4708513
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4711450
4712917
29073.4634450
2908 3.4635944 1
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'2912 3.4641914!
|2913 ! 3.4683405!
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295713.
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2916 3.4647875 29663 4
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'2923 3.4658288;;2973 ! 3.4731949
!2924;3.4659774 ! |2974'3 4733410
'2925'3.4661259| 2975,3.4734870
2871 3.4580332
2872 3.8581844
2873 3.4583356
2874 3.4584868
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3 4587889 ;2926 3 4662743
3.4589399 2927| 3.4664227
3 4590908 29C.3 3.4665711
3.4592417 : |2929 , 3.4667194
2976
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2978
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3.4593925|!2930 3.4668676! 2980
3.4595433
3 4596940
3:4598446
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34601458
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3.4604468
3.4605972
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3 4608978
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28973.^
28983
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2931 346701581
129323.4671640
j2933 ! 3.4673121
|2934'3.4674601
2935'3.467608l
J29363.4677561
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293913.4681996
! 2940'3.468347;
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3.4737788
3.4739247
3.4740705
3.4742163
2981
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4746533
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4768316
4769765
477121
ARITHMETIC*.
84$
3011
3012
3013
3014
3015
3016
3017
3018
3019
3020
"No. JLo£. No.l Los*.
3001
3002
3003
3004
3005
3006
1007
3008
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3010
3 4787108
3 4788550
3.4789991
3.4791432
3.4792873
3021
3022
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10253.
3026
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: 3030 3
3031
3032
3033
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3 4772260
3.4774107
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3.4776999
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3.4779890
3.4781334
3.4782778
3.4784222
34785665
3 4794313
3.4795753
3.4797192
4798631
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34801507
3.4802945
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3.4805818
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4808689
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4811^59
4812993
4814426
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4818724
4820156
4821587
3051
3.4844422
30523.4845845
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3054J3 4848690
30553.4^50112
30563.4851533
30573.4853954
30583.4854375
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3060'3.4857214
31013.4915018
1023.4916418
3103 3 4917818
31043.4919217
31053 4920616
3061
3062
3063
3.4858633
3.4860052
3.4861470
130643.4862888
1396534864305
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068
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3070
3071
3072
1073
3074
3075:
30363.
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3039
1040
3.4865722
3.4867138
3.4868554
3.4869969
3.4871384
3.4872798
3.4874212
4875626
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3.4878451'
3076
077
3078
3479
3080
3081
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3041
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3 4831592
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31073.4923413
31083.4924810
31093.4926207
31103.4927604
31113.4.929000
31123.4930396
3113 3.4931791;
31143.4933186
3115 3.4934581
3116
117:
3118
119
3120
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1.4896773-!
1.4898179!
1.4899585'
4833020
4834446
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4840150
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3092
309.
3094
3095
3096
3097
3098
3099
3100
1.4900990 :
1.4902395!
14903799;
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149066071
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4909412.
.4910814'
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No., Loe.
3151|3.49iJ4484
3152i 3.4985862
31.53 ! 3.498724
3154'3498861
31553 4989994
3156 3 4991370
3157*.°4992746
3158 3.4994121
3159 3.4995496
3160 3 4996871
3.4935974
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3.4938761
3.4940154
3.4941546
312113.4942938
31223.4944329
31233.4945720
31243,4947110
312534948500
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31293.
130J3.
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4951279
4952667
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l^o<.
31613 499*245
1623.4999696
163 3 5000992
1643 5002365 1
3165 3 500373;
No. Log 1 .
li^Ol
3202'
3203,'
3204
3205
O.D052857
3.5054218
3.5055569
3.5056925
3 505828c
3.5059635
3 5060990
3 5062344
3.5063697
3.5065051
3166 3.5UU5109
31673 5006481
3168 3.5007852
3169 3.5009222
3170 3.5010593
31713.5011962
3172'35013332
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J3174 3.5016069
3175 3.5017437
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4977587
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314913 4981727
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3176
177
178
3179
3180
3181
3182
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3184
3185
|3186
3187
4188
3189
3190
3.5018805
3.5020172
3.5021539
35022905
3.5024271
3,502563;
3.5027O02
3.5028366
3.5029731
3.5031094
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3210 1
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321213 5067755
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3214 3.507045S
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16 3.507316C
3217
3218
3219
3220
3.9074511
3,5075860
3 507721C
3 507855S
3221
3222
3223
3224
3225
3 5079907
3 5081255
3508260J
3.5083950
3.508529;
3226
3227
3229
3230
3231
3233
3234
3235
3.5032458
3.5033821
3.5035183
3.5Q36545
3 5037907
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3.5086644
3.5087990
3228 3.5089335
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3.5092025
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8244 35110608
3245 3.5US14?
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3247 3.51 14823
3248)3.5116160'
3249[3 51174PT
3?50"3.5!1J!834?
1
220
LOGARITHMIC*
No.
Log.
3551 3.5120170
325255121505
3253 3 5122841 1
3254 ! 3.5124175'
3255 3.512551Q
3256 3.5126844
3257 3.5128178
3258 3-5129512
325£ 3.5130844'
3260 3.5132176'
No.| Lop.
33 jl 3.5186455
33023^187771
33033.5189086
33043.5190400
3305 3 5191715
33063.519302S
33073.5194342
33083 9195655
33093.5196968
33103.5198280
3261 & 5133508
32623513-1840'
3263 3.51361711
3264 3 5137502;
3265 3.513883 21
3266 3.5140162J
3267,3.5141491
3268 3.5142820
3269 3.5144149
3311 3 5199592
33123.5200903
3313 3.5202214
33143.5203525
33153.5204835
33163.5206145
3317 35207455
3318 3.5208764
.3319 3.5210073
3270 3.5145478! 3320 3.521 1381
3271 3 5146805
3272 35148133
32733.5149460
32743.5150787
3275 35152113
32763.5153439'
3277 3.5154764*
8278
3279
3.5156089
3.5157414
32803.5158738
C28l|3.5160062
5161386
3.5162709.
3.5164031J
3.5165354
3282 3.
3283
3284
3285
3286
3287
3283
3289
3290
3291
3292
3293
3294
3295
3296
3297
3298
3326*3.5219222
33273.5220528
33283.5221833
33293.5223138
3330)85224442
33313.5225786
3.5166676
3.5167997
3.5169318
3.5170639
3 5171959
3.5173279
35174J98
3.5175917
3 5177236
3.5178554
3.5179872
35181189
3.5182507
3299;3.5183823
3300 351 851 39
3321 3.5212689
3322 3 5213996
3323 3.5215303
33243.5216610
3325-3.5217916
1332
3.5227050
3334
3335
33333.5228353
3.5229656
3.5230958
3336
3337
3338
3339
3340
3.5232260
3.5233562
5234863
3.5236164
35237465
8341
3342
3343
3344
3345
3346
3347
3348
334?
3350
,No.
,3351
'3352,
'3353
[ 3354i3.
3355
Log.
3.5251744
35253040
8.5254336
5255631
5256925
No.
3401
3402,«.
3403*3.
3404
3405
Lo^.
5316066
3.5317343
5318619
3.5319896
3.53*1171
33563 5258220 3406:3.5322446
3357 3.5259513 3407 3.5323721
3358 3.5260807 34083.5324996
!3S59 3.5262100 3409:3.5326270
13360 3.5263393]3410 3.5327544
ii *<
!336l|3.5264685; 3411J3.5328817
3362(3.52659771
3363,3 5267269
3364 3.5268560 1
3365,3.5269851
3412,3.5330090
34133.5331o53
3414'3.5332635
341535333907
33663.5271141
3367135272431
3368'S 5273721
3369.3.527501 Oj
3370,3.5276299
3416,3.5335179
34173.5336450
3418 3.5837721
3419 3 5338991
3420 3.5340261
3371,3.5277588'
33723.5278876!
3373 , 3.5280163.
3374'3.5281451i
3375 , 3.528273 8;
(3376 3.5284024
■337713.5285311'
3378'3.5286596!
3379 3.5287882
3380 3.5289167
3421
3422
3423
3424
3425
3426J3 5347874
3.5349141
3.5350408
3.5351675
3.5352941
3427
3428
3429
3430
33813.5290452
3382'3.5291736H:
3383 3.5293020 :
3384 3.5293304 I
13385:3.5295587 !
3.5238765!
3.5240064!
3.5241364;
3.5242663!
3 5243961
3.5245259
3.5246557
3.5247854
3.5249151
5250448
3386 3.5296870
3387J3.5298152,
3388 3.5299434 !
33893.5300716'
3390 3.5301997
33913.5303278
3392,3.5304553
33933 5305839
33943.5307118
3395, 3.530839 8
33963.5309673
3397j3.5310955
3398.3 5312234
3399 3.5313512
3400 ! 3.5314789
3431
3432
3483
3434
3435
3436
3437
3438
3439
3440
3441
3442
3443
3446
3447
3448
3449
3469
3470
3.5354207
3.5355473
3.5356738
3.5358003
3.5359267
3482
3483
3484
8485
3.5360532 3486
3.5361795 3487
3.5363059
3.5364322
3.5365584
3.5366847
3.5368109
S 5369370
34443.5370631
3445 3.5371892
5373153 13496
3
3.5374413
3.5375673
3.5376932
345035378191
3451
3452
3453
3454
3455
3456
3457
3458
3459
3460
3461
3462
3463
3464
3465
No.
3.5379450
3.5380708
3.5381966
3,5383223
3.5384481
8.5385737
3.5386994
3.5388250
3.5389500
3.5390761
3.5392016
3.5393271 *
3.5394525
3.5395779
3.5397032
3466 3.5398286
346735399538
34683.5400791
3.5402043
35403295
3 5341531 3471 3.5404546
3.5342800 3472 3 5405797
3:5344069 3473 3.5407048
3.5345338 3474 3.5408298
35346606 3475 3.5409548
3476
3477
3479
3480
3488
3489
3490
3491
3492
3493
3495
3497
3498
Log:.
3.5410797
3.5412047
34783.5413296
3.5414544
3.5415792
348J&5417040
3.5418288
3.5419535
3.5420781
35422028
3.5423274
35424519
3.5425765
3.5427010
35428^54
3.5429498
3.5430742
3.5431986
34943.5433229
3.5434472
3.5435714
3.5436956
35438198
3499J3 5439439
35QQI3 543068gl
ARITHMETICS
22l
Log. t|No,, Log.
No.]
35^1 3.5441921 3^513.55035071
3502 3.5443161 '3552 3.5504730
3503 3.5444401 [3553 3.5505952
35043 5445641 '35543.5507174
35051 3.5446880 '3555 3.5508396
3506*3.5448119 3556 3.5509618
3507 3.5449358 '3557 3.5510839
3508 3.545Q596 3558 3.5512059
3.5451834 3559 3.5513280
3.5453071 3560 3.5514500
&.5454o08 3561 3.5515720
No.
Log.
3.5564231
3.5565437
3.5566643
3.5567848
35569053
3509
3510
3601
3602
3603
3604
£05
36063.5570*57
2607 3.5571461
3608 3.5572665
3609 3.5573869
36103.5575072
3511
351213.5455545 ;3562 3 5516939
3513J3.5456781 ! 3563 3.5518158
3514'3.5458018 '3564 3.5519377,
51513.5459253 13565 3.5520595
No. I Log ._ . , No.
3651 3.56241 18l|370i
3652 3,5625308' 3702
1653 3.5626497^3703
36543.562768513704
655 3 5628874 13705
>656 3.5630002
611
3612
361
3614
3615
706-
657)3.5631250 J3707
658 3^632437 3708
3659 3.56336M 13709
6603.5631211*3710
L og. |
3.5683191
3.5684364
3.5685537
3.5686710
3.5687882
3516 ! .3.5460489 t !3566 35521813
3517 3.5461724 '3567 3.5523031
351813.5462958; 35683.5524248J
3519|3.5464193 33569 3.5525465!
3520 3.546542 7 1 3570 ' 3.5526S8 2|
3521 3.5466660JI3571 3.5527899J
36i6
3617
3618
3.55/0275
3.5577477
3.5573686
3.5572881
3.5581033
3 5582284
3.5583485
3 5584636
619,3.5585886
3661
3662
366:
3664
3665
3660
667
3668
,620
1621
3622
3.5587086
3.5689054
3.5690226
3.5691397
3.5692568
3.5693739
3.5635997'3711
3.5637183 ! 3712
3.5638369 3713
3.5639555 3714
3.5640740 ! 5715
3.5694910
.5696030
36697249
3.5698419
3.5699588
o.5ofcti285
3.5589484
3.5590683 1
55918821
3522|3.5467894 ; J3572 3.5529115!
35233.5469126:3573 3.5531030113623
3524 3.5470359, 3574 3.5531545 3624
3525 3.5471591 3575' 3.5j32760 '!3625 3 5593080 ',
5526 3.5472823. 3576 3.5533975J
3527 3.5474055 3577|3 55351 89i
3528 3.5475286(135783.5536403
3529 3.5476517 3579 3 5537617
3530*1 3.5477747 1 3580 1 3.5538830 1
3531 3.5478977 J3581 (3.5540043
o.5641925j37l6i.>.o7UU7 57
3.56-13ld9; 371713.5701926
_ __, 3.5644293' 1 3718 3.5703094
3669 3.5645477! 3719 3.5704262
367ol 3.564666 l! : 3720' " 5705429
00? 1 13.504? 844 ! : 67~7i |T370659?'
oO*o|o.5594278
3627 3 5595476,
3532 3.5480207p582|3.5541256
3533 3.5481436 3583|3 5542468
35343.5482665 35843.5543680
3535 3.5483894'
35363.5485123!
3585 ! 3.5544892
353713.5486351
35383.5487578
35393.5488806
2672 3 5649027
3673 3.5650209
3674^3.5651392
3675 3.5652573
00/ o!3 5653755
3677 3.5654936
3628)3.5596673' 36783.5656117
3629 3.5597870.
3630 3.5599066'j
,3631
3632
3633
3634
3635
3586.3.5546103
358713.5547314
3588 ! 3.5548524
,35893.5549735
3540 3.5490033! 3590 3 555Q944
354113.5491259! 3591^ 5552154
~ 542,3.54924861 3592 3.5553363
35433.54997121
3544 ! 3.5494937
3545' 3.5496162 i
3593 3.5554572
35943.5555781
3595'3.5556989
35463.54973871
3547j3.5498612|
3548 3.5499836
3636
3637
3638
3639
3640
5641
3642
.5600262
3.5601458
3.5602654
3.5603849
J5605044
3.5606239
3.560743;
3.5608627
3.5609821
; 561101 4 \3690
3722 3.5707764
3723
3724
3725
3726
3727
3728
3.5708930
3.5710097
3. 5711263
5712429
3.5713598
3.5714759
5715924
35717088
3679 3.5657297 3729
36803 .565847 8 3730 ~^ JZZ _ z:zr
3681 3.5659658 37313.5718252
3682 3.5660838 3732 3.5719416
3683 3.5662017 '3733 3.5720580
3684 3.5663196 3734 3.5721743
i3685j3.5664S75 ! 3735 3.5722906
6066 o.506oo5o\jo736|3 5724169
3687 3.5666731 1 3737 3.5725231
3688 3.5667909! 3738 3.5726393
3689
3.5669087i!37S9|3.5727554
3.5670264 ! '3740 3.5728716
3.5612207 |o6yi|3.5671440,3741o\5729877
3.5613399 3692
1643 3 5614592
3644 3.5615784
3645 3.5616975
3596 3.5558197 U646 3.5618167
3597,3.5559404 3647 3.5619358
3598 3.5560612 3648 3.5620548
369:
8694
,3695
3696
13697
3698
3699
3549 3 5501060 3599 3.5561818 3649 3.5621739
3 55013.5502284113600 3.5563025'13650l3 5622 929.l3700
29
3.5672617i!3742 3.5731098
3.567«793i ! 3743 35732198
3.5674969; 3744 3.5733358
5676144 ! ! 3745 3.573451 P
0.66779*0 S746 3"5?35S7£
3.5678495:' 3747 3 573683T
3.5679559|374* 1 } 5737996
3.5670843: 3741 h. 5739 154
3.56720171 3750!3.57403li:
i
22S
LOGARITIIMICK
37513.
37523
37533
375^3.
37553.
i\i>[ Lotf.
37563
3757:3.
3758'3
3759)3.
37603
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3762J3
37633
3764 3
3765:3.
3766|3
3767 3
37683
376913
37703.
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3772
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3774
3775
3776
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3778
3779
3780
3781
3782
3783
3784
3785
.6741471
5742628!
5743786
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5746091
5747$5(
57484U
574956h
575072.5
5751878
5753099
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5756496
5757650
ooi/l 3.579#97<
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3.5801263
3 580240^
3 5803547
£5804688
380:
3804
3805
'3806
3807
3808
3809
3810
38li!oj«10389
3.5805821 3857 3.5862496
3 5806969
3 5808110
3 5809250
5758803
5759^56
.5761109
5762261''
.3763414
5764565
5765717
5766868
.5768019
.\«i.» i^^.
'3Hd1o.o&5o735
138523.5856863
•3853 3-5857990
38543 5^59117
3855 3J860244
ooob o. 5861 370
3901
3902
3903
3 6911760
35912873'
3.5913986
3904 , 3.5915098
3905J3 5916210
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3907^3-5918434
3908 3.5919546
3909 3.5920657
3910J 3 5921768
^Sol o^Loo^6i!39n'3 5922878
381213 5811529(!3862 3.5868123 ]39123 5923988
.38583 5863622.
J3859 3 5864748:
^60 3 58658731
l^og. h^u.j Loj>.
3951 OO967070
'3952 3 5968169
]3953 3 5969268
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3955J3 5971465
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35973661
3.5974758
3.5975855
3 5976952
3863 3.5869247 13913:3.5925098.,
3864 3.5870371 ! 3914 3.5926208
13865 S_5871495|i3915!^5927318
13866 5.5872618| 1 ;*16 ! 3*5928427
|3867 : 3.587S743| i 39l7 , 3 5929536
13868 3 5874863 39183.5930644
3813 35812668
38U3.581S807
3815 3 581494 5
J3816J.5816084
! 3817'3.5817222
,3818 3.5818359,-- . „ tnrt ,_
*3819 35819497 3869 3.5875987 |j39 19 3.5931753
3822'3.5822907 3872 3.5879751
3823 3.5824043 ! 3873-3.5880475
3824 3.5825179 13874 3.5881596
5769170 38253 5826314 J3875 3.5882717
.>,86
3787
37883
3789
3790
3791
3792
3791
3794
3795
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3797
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3799
3800
8203 5820634 J3870 3.5877110
3821 6o»21770
j33n;3.5878232i!3921 3 5933968
J3920 3 5932861
.5770320 38263.5827450
.5771470 38273.5828585
.5772620 38283.5829719
5778769 38293.5830854
5774918 3830 3.5831988 '3880 3.5888317 ;
5776097 o<\>1 3.5833122
5777215 38323.5834255
5778363 38333.5835388
5779511 38343.5836521
5780659 38353.5837654
5781806 otto6 3.5838786
5782958 38373 5839918
•5784100 38383.5841050
.5785246 38393 5842181
.5786392 38403.5843312
l3925 ; 35938397
;3o; (i3 5883«38,
138773.5884958
'38783.5886078
138793.5887198;
.5/87538 3841 3.5844443
5788683 38423.5845574
5789823 38433.5846704
579397S! 38443.5847834
792118133453.5848963
5799262: 38463.5850093
.5794406 f 3847 3.5851 222
5795550, 38483.5852351
.5796693 38493.5855479
5797836 38503.5854607
3926 o.o K Jb^o03
927 3.5940509
3928/3 5941715
3929 3.5942820
930*3 5943926 3<)80 35998831
; 3881 3.586943d
38823.5890555
! 3883 3.5891674
'38843.5892792
3885 3.5893910
3bb63.589502tf
38873.5896145
38883.5897263
38893 5898379
389035899496
'3891 o.oy00G12
3892^.5901726
3893 3.5902844
38943.5903959
3895 3.59050 75
3896 o^uol89
38973.5907304
38983 5908418
38993 5909582
39003.5910646
3 5979145
. 3.5980241,
13964 35981336
3965 3 698243%
3966
>967
3968
;969
o.^b3327
3.5984692
3.5985717
._, 3.5986811
3970 3.598 7905
^71
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3973
(3922,3.5935076
8923'35936083
3924t3.5937290 39743.5992279
o.o*,t&999
3 5990092
3.5991186
3975
3976
3977
3978
3979
3 5993371
3.5994464
3.5995556
3.5996648
35997739
o93lM-^45030
3932'3.5946135
3933-3 5947239
8934*3.5948344
3935? 3 5949447
3936i3.5950551
3937|3 5951654
3938'3.5952757,
3939;35953860
3940 3 595496 2
3941 o 5956064
3942 3.5957166
3943 35958268
3944 3.5959369
3945 3^960470
3946 3 5961571
39473.5962671
3948 3.5963771
39493.5964871
39503.5965971
,i>joi o.jb>99922
|3982|36901013
3983J3.6002103
3984'36003193
3985* 3 6004283
3986'3.6005o/3
3987|3.6006462
3988i3.600755\
3989 ! 3 6008640
399036009729
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39923.6011905
9933.6012993
39943.6014088
399*3 6015168
3 ( >9bo.uv/ 10255
3997'3 6017341
39983.6018428
3999 3.6019514
40003 6020600
ARITHMETICS
$23
400 i
4002
4003
4004
4005
;no.
Lo»{. J
5.602 lotft
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3.6023857
3.6024941
3.6026025
3 6076694
4055 J 6077766
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4055 3-6079909
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4061 3 60863 JO 4111
4062 3.6087399* 4112 36140531
4063 3.5088468 '4113 36141587
40643.6089537 41143 6142643
4065|3 6090605 41153.6143698
40l63.'J03fyJ7,
40173 6039018
40183.6040099
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402036042261
4066'
4067
4068
4069
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14022 3 6044421
4023 5.6045500
40243.6046580
4025 3 6047659
40:40 3.6048738
40273.6049816
40283.6050895
40293.6051973-
403036053050
4031S6054128
4032 3.6055205
4033 3.6056282
40343.6057359
4035 3.6058435
4^56 36029512
4037 3.6060587!
40383.6061663'
4039 3.6062739
40403 6063814
40-* i 3.60d4«ay 1
40423 6065963
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410,
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3.61^2074
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4107
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3.6106602
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41233.6152193
4124
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4078
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4082 3.6108730
408336109794
4084'3.61 10857
408513.6111921
4128
4129
1.61-53187
1.6154240
4126 3.6155292
4127 3.6156345
6186755
6187800
6188845
6189889
6190933
6191977
6193021
.6194064
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.6196150
4206 3.0238693
42075.6239725
4208 3.6240757
42093.6241789
42103.6242821
3.6144754 4 1 6613.0197 193
3.6145809 4167|3.6198235
3J5146863 I416S3.6199277
: 4169 5 6?00319
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, 4174'3.6205524
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3.6157397141783.6209684
3 61584491
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4134*3 6163705,
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4087 3.6114046 J4137J3.61 66855:
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1197 3
4198 3.
4200:
.6228354 |
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6230424
633145S
6232493
No. I Log.
4201.3.6233527
4202 3.6234560
4203 3 6235594
42043.6236627
4205 3 6237660
42113.5245852
421236244884
4215 5.6245915
J4214 5.6245945
42153 6247976
;4216 5o249006
142173 6250056
14218 5.6251066
'4219 5.6252095
42205 6253125
4221 5.0254154
42225.6255182
4225S.62562I1
J4224 56257239
J4225 56258267
42263.
42273,
4228 3
4229 3.
4230 3
6259295
6260322
6261350
6262377
6263404
4231
4232
4235
4234
4235
6264430
6265457
6266483
,6267509
,6268534
42363 6269560
4237 3.6270585
;4238 , S6271610
|4239!3.6272634
4240'3,627S659
,424 1 ! 3. 6274685
12423.6275707
142433 6276730
i4244 3 6277754
4245 3 6278777
42463.0279800
•424736280820
14248 36281843
42493.6282865
142503.6283887
tu
LOGARITHMICK
4Jd1'3 6284911
6285933
6 86954
6087975!
^2SS996*
42J23
4253
4254
4255
425o<3
42573
4253*3
4259|3
4260:3
42bl!3
4262|3
4263,J>
4264|3
4265J3
N T o.| Log.
,Ncj
4301J3
43U2|3
4303'3
43043
4305!3
\ k«g || No. Lo^.
O^jjaloj43u63
629l03r!4307 r i3
629^05^43083
6293076 U309'3
629409t>!;4310'3
6335694 4351 3
63367<M , !43523
6337713|43533
63387231435412
6339732:14355 3
6j40/4^|4o563
6341749; |4357 3
6342757! |4358 3
6343765; J4359 3
6344773 ]4360 3
4266P
42673
42683
42693
42703
t>4/511jj:4311;3
629bl34 14312 ! 3
629*153 ;4313'3
6298172 *4314 ! 3
6299190f.4315 , 3
431t>j3
4317&
4318;3
143193
143203
6o*57bU ;43bl 3
6346788 4362 3
6347795 4363 3
6348801' 4364 3
63498084365 3
6385891
6386889
6387887
6388884
6389882
63908? 9
6391876
6392872
6393869
6394865
0395861
6396857
6397852
6398847
6399842
No.| Log.
44Ul;3 6435d1*
1402 3 6436500
4403
44043 6438473
4405
4406
4407
4408
4409
14410
6300^09
6301226
6302244
6303222
6304279
4271|3
42T23
4273p
42743
42753
6^05290
6306312
6307329
6308345
6309361
4276,3
42772
42783
427913
4280|3
631037/
6311393
6312408
6313423
6314438
4281|3
4282'3
4283'3
4284 ! 3
4285'3
6315452
6316467
6317481
6318495
6319508
4'itf 6 3
4287:3
42883
4289^3
42903
6J2J522
6321535
6322548
6323560
6324573
42913
42923
42933
4294*3
4295;3
6325585
6326597
6327609
632*620
63295.V2
42963 6330643
429713 6331654
4298'3 6332664
4299:3 633367*
4300'3 63346*
6350814
6351820^
6352826'
6353832!
6354837
4321
4322
4323
43243
63558431
6356848
6357852 f
63588571
63598611
43263
43273
43283
43293
43303
6360805
6361869
6362873
6363876
6364879
'433113
143323
143333
, , 4334 , 3
J4335I3
6365882
6366884
6367887
6368889
6363891
43661,
4367
4368
4369
4370
4371
4372
4375
43743
43753
4376
4377
4378!
4379
43803
4381
4382
4383
4384
4385
4336|3 6370393*43863
433713 6371894 J4387
4338J3 6372895 '4388
4339 3 6373897 '4389 3
4340|3 6374897:4390 3
4341
4342
4343
4344
4345
43463
4347
,43483
434913
14350
6375898
6376898
6377898
6378898
6379898
•6380897
6381896
6382895
6383894
638489:
;4391
14392
4393
4394;
4395
4396
4397
4398
4399
4400 1 .
64UU837J
6401832|
6402826!
6403820J
6404814
4416'3
44173
44183
44193
4420J3
6405808
6406802
6407795
6408788
6409781
6410773
6411765
6412758
6413749
6414741
64157331
6416724
6417715
6418705
6419696
6420686
6421676
6422666
6423656
6424645
6425634
6426623
6427612
6428601
6429589
6430577!
6431505
6432552
6433540
4411
4412
4413
44143
4415!
3 643748;
, No. '
4451
14452
445.
6139459
6440445
6441431
6442416
6443401
6444386
Log-
3 6484576
3 6485552
3 6486527
4454 3 6487502
44553 6488477
6445371
6446955
6447399
6448323
6449307]
44613
4462*3
4463 ; 3
4464'3
4465',3
6450291
6451274
6452257
6453240
6454223
4421|3
44223
44233
44243
44253
6455205
6456187
6457169
6458151
6459133
44263
44273
44283
4429(3
4430*3
4431-3"
44323
4433 3
4434 ! 3
4435J3
6460114
6461095
6462076
6463057
6464037
4466;3
44673
4468*3
44693
44713
44723
4473
44743
4475
6465018
6465998 1
6466917^
6467957
64689361
44363
4437J3
44383
443913
4440 3
44413
44423
44433
44443
44453
6469915!
6470894!
6471873J
6472851
6473830
6474808
6475785
6476763
6477741
6478718
44463
444?;3
44483
4449 3
3 643452744 50 3
6479695'
6480671!
6481648
6482624
6483600!
44563 6489452
44573 6490426
44583 6491401
4459 3 6492375
44603 6493349
6494332
6495296
6496269
6497242
6498215
6499187
6500160
6501132
6502104
6503075
44763 6509901
44773 6509871
44783 6510841
6511811
44803 6512780
4481
4482
4483
4489
4490
4498
4499
4500
6504047
6505018
6505989
6506960
6507930
3 6513749
3 6514719
3 6515687
44843 6516656
44853 6517624
44863 6518593
44873 6519561
44883 6520528
3 6521496
3 6522463
4491 3 6523451
4492 3 6524397
4493 3 6525364
44943 652633V
4495 3 6527297
4496 3 6528263
4497 3 6529229
3 6530195
3 6531160
3 6532125!
ARITHMETICK.
2ft
4501
4502
4503
4504
4505
450613.6547912
450*3.6538876
45083.6539839
36540802
3.6541765
4509
4510
4511 3.6542726
45123.6543691
451313.6544653
4514 , 3.6545616!
45153,6546578'
4516<3.6547o39!
4517&654850li
45183.6549462:
4519,3.6550423
45203.6551384
No
3.6533090
3.6534055
3 6535019
3.6535984
3.6536948
Lo£.-
3.6695359
3.6596310
45683.6597261
,6598212
3.6599162
4569
4570
4121(3.6552345
45223.6553306
45233.6554266
4524'3.6555226
45253.6556186
45263.6557145
4527J3.6558105
452836559064'
4529,3.6560023!
45303.6560982'
45ol3 6561^41
4532 3 6562899
4533-36563857.
4534 f 3 6564815
453536565773
4536 3.
4537|3
4538.3
45393,
4540 f 3
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6567688
6568645
6569002
6570559
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4545 3.
6571515
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6573427
6574383
.6575339
45463.
4547,3
4543,3
45493
4550'3
,65762y4i
6577250,'
6578205!
6579159
6580114
No. Lop:.
4551
4552
455^
4554
4555
3.65810*
3.6582023|
3.6582977
3.6583930J
3.6584884!
4556
4557
4558
4559
4560
3.6585837
.6586790
3.6587743
3.6588696
3.6589648
4561
4562
4563
4564
4565
3.6590601
6591553
3.6592505
3.6593456.
3.6594403
4566
4567
4571
4572
4573
4574
4575
4577
4578
4579
4580
4581
4582
4583
4584
4585
4586
4587
4588
4589
4590
4601
4602
4603
4604)3
4605
4606
4607
4608
4609
4610
4611
4612
461,
4614
4615
3.6600112
3.6601062
3-6602012
3.6602962
3.6603911
4616
4617
4618
4619
4620
4621
4622
4623
4624
4576 3 66048OU,
6605809i
3.6606758
3.6607706;
3.6608655;
3.6609603
3.6610551
3.6611499i
36612446
3.6613393
3.6614341
3.6615287,
6616234
3.6617181
3.6618127
4591j3.6ol9u/o
4592j3.662001y
4193J3.6620964,
45943.6621910
4595I3.6622855 1
45ybj3.6623800
45975.562*745!
3.6625690
3.6626634
3.662757&
4598
4599
4600
No,
3.6637951!
3.6638893
3.6639835
3.6640776;
3.6641717
3.6642658
1.6643599'
3 6644539!
3.6645480 1
36646420'
4625 3
4627
4628
4629
4630
4o31
4632
463:
4634
4635
4636 3~
4637
4638
4639
4640
Loir. "| f No. | Log.
3.6628522
3.6629466 1
3.6630410J
,6631353|
3.6632296
4651 3.6675463
46523.6676397
4653 3.6677331
46543.6678264
4655 3.6679197
3 6633239 4666 3.6680130
3.6634182 46573 6681062
3.6635125 4658 3 6681995
3 6636 J67i 4659 3.668292
3 6637009 4660 3.6683859
3.6647360'
3.6648299,
3.6649239!
3.6650178
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4626 3.6652056
3.6652995!
3.6653934]
3-6654872j
3.6655810'
3 6656748;
36657686
3.6658623
.6659560
3.6660497,
6661434
6662371!
3 66633071
3 6664244;
6665180
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6667957j
66679871
6668922
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.6670792'
3#671727
3.6^72661!
3.6673595'
3 6674530-
46613
4662 3.
4663 3.
4664 3,
46653
No-
4701
4702
4703
4704'
'4705
Lotf-
3.O721903
3.6722826
3.6723750
3 6724673
3 6725596
47 06
.4707'
'4708
4709
4710,
6684, Jyl*
6685723
6686654*
6687585
6688316
46663
4,6673,
46683,
46693
46703,
4/11!
4712 1
4713 1
4714'
4715 ;
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66903? 8
6691308,
6692239 ;
66931691
4/16
4717»
4718'
4719 1
4720 1
46713 6694099
46723.6695028 !
46743.6695958!
46743.6696887;
4675 3 669781 6,
46763.669874'5
46773.6699674!
46785.6700602
46793.6701530
4680 3.6702459
4/21
4722
4723!
4724 !
4725:
4681 3.6703386
4682 3.6704314
4683 3 6705242
4684'3.6706169
4685*3.6707096
4686 3.6708023
4687'3.670S950
46883.6709876
4689'3.6710802
46903 6711728'
469 1,3.67 126o4l
4692
4693
4694
4695
6713589
3.6714506^
3.671543-11
3 6716356'
4696
4697
4698
4699
4700
3.671/281!
3.6718206'
3 6719130'
3.67200541
367209791
4726i
4727J
4728
4729!
47301
4/31
4732
4733
4734
4735
o o/265ii.'
3.6727442
3 6728365 .
3.6729287
3.6730209
3.6/31131
3 6732053
3 6732974
3.6733896
3 673481 7
o.o/Jj/38
3 6736659
3.6737579
3.6738500
36739420
3.6740J40
36741260
3.6742179
3.6743099
3 6744018
J.6744937
3.6745856
3.6746775
3.6747693
3.6748611
3.674^529
13.6750447
3 6751365
13.6752283
l3.fi/53200
4/jb
4737
4738'
47391
4740;
o.u/ 04117
: 3.67 55034
3 6755951
3.6756867
6757783
4741'
4742;
4743;
4744i
4745 1
4746>
4747!
4748
4749
4750
o.o? 58700
3.6759615
3.6760531
3.6761447
367*2362
a.o/t)3277
3.6764192
3.6765107
3.6766022
36766936
526
LOGARITHMICK
No. L
|N«>., Leu*
47oi 3.0/orojj
47523.6768564
4753 { 3 6769678
4754 3.6770592
4755'3.6771.505
<i 8 Ji 3.68133 1/
4802 3.6814222'
i4803'3.6815126
! 4304'3.68t6010
48053.6816934
47*>o 2,
4757%
47583
4759 \S.
4760 3.
4701 3"
4762 3,
4763;3.
4764'3.
476.^3.
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6774244 1 4808,3
677515/;
6776070'
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.6777894
,6778806
6779718
,6780529
4/uo3.o7t>io4u
4767,3.6782452
47683.6783362
4769:3.6734273
47703.6785184
4/71 '3.67*01/**
4772J3.5787004
4773J3.6787914
47743.6788824
4775'3.67897S4'
47/63.57yoo4j
4777|3.6791552
4778,3.6792461
47803.5794279
480$:
4810,3
48113"
48123.
481313,
48143
4815 3.
4851 b 6858313
4852*3 6859208
4853 3.68601C3
4854 ;3 .6860998
4855 3.686189 2
6817838 !48o6:3.6802/«7
.6818741 4857'3.68f>3681{
6819645! 4858*3 6864575J
6820548 1 48&I3.6865469 1
6821451' 4860'3 6866363
.O8^oo4
6823256
.6824159,
6825061,
6825963
4862
4863
4864
486
481o3.6826865
48173 6827766
4818 3.6828668
48193.6829569
48203.6830470
4821 3.6831371
48223.6832272
4823 3.6833173
48243.5834073
48253.6834973
4861 '3-6867256
4866
4867
4868
4869
4870
4826 3.6835873
482736836773
148283.6837673
**/ f o,j.os y&kOl fbojo j.cvwfo/**
4779 3.6793370 14829 3.6838572
'48303.6839471
4871 3.68761811 4y21;3.by20534|!4971
4872 3.6877072!i 4922 3.6921416! 4972
487S 3.6877964 4923 3.6922298! 4973
4874 3.6878855, 49243.6923180 4974
4875 3.68797461 4925 3.6924062 '4975
487636880637
3.6881528
3.6882418
3.6833308
36884198
4877
4878
4879
4880
4781 3.679518/1.4831 3.6840370 14881
4782 3.6796096 4832 3 6841269 4882
4783 3.6797004 4833 3.6842168 '488.
4784 3 6797912 '4834 3 6843066 ,4884
4785 3.679981 9) j 4835 3.6843066 [ 4885
4786 3 6799/27! 14836 3.6844863 ,4886
4787 3.6800634 4837 3.6845761 4887
47883 6801541 4838 3 6846659 '4888
4789 3.6802448 !4839 3.6847555 '4889
4790 3.6803355 484013.6848454 4890
4/yi 3 68042o2
1792 3.6805168
47933.6806074
47943 6809680,
1795 3.6807836!
4796 3 6808792
47973.6809692
47983 6810602
47993.6811507
4800 3.6822412
4841l3.68493ol
48423 6850248
4843
4844
4845
4846
4847
4849
4850
3 6851145
3.6852041 1
3.6852938
4891
4892
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3 6853834;
3.6854730;
'48483.6855626^
3 6856522J
3.6357417 1
I
48'
48<
4898
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4900
3 6868150
6869043
3.6869936
3 6870828
3.6871721
3.6872613
3.6873506
3.6874398
5.68'5290
No. f L,og:. i!No| Lotr.
49U1 3.by02847U951 3.6946929
4902 3.6903733 Ij4952'3.6947806
4903 3.6904019 |4953 3.6948683
49043.6905505
4905 3.6906390
49543 6949500
f4955 3 6950437
4906 3.090727 j, ,4956 o.
4907:3'6908161;;495713.
49083 6909046:4958 3.
4909 5.-6909930! 14959,3.
4910 3
49fcl|3.
4962J3.
6951313
6952189
6953065
.6953941
6954817
5.69U699j
L6912584 1
5.6913468!
149633,
3.6885088
3.6885978
3.6886867
3.6887757
3.6888646,
3.6889535
3.6890423
3.6891312
3-6892200
36393089
3.6893977
.6894864
3.6895752
3.6896640
36897527
3 6898414
3.6899301
36900188
16901074
3 6901961
4911 3,
49123
4913 3 ,ww. i -«w W w.
4914 3.69143521 14964 !3.
4915 3.691523 5J!4965'3
49163.6916119
4917'36917002
49183.6917885
6955692
6956568
6957443
.6958318
6959193
J4966J3 6960067
'4967I3-6960942
l4968'3.6961816
4919 3.6918768 J4969J3.6962690
49203.6919651 l 4970!3.6963564
4926 3
49273.
4928i3,
4929 3.6927588,(4979
4930
3.6964438
3.6965311
3 6966185
3.6967058
3 6967931
.6924944:14976
,6925826! 4977
6926707(4978
5.6928469,4980
J4931 3.6929350 J4981
4932 3693023i;'4982
4933 3.693UUIJ4983
49343.69319914984
[4935 3 6932872 14985,3 6976652
>936
|4937
4938
4939
4940
4941
4942
14943
4946
4947
6968804
36969676
3.6970549
3.6971421
3. 697229 3
3 6973165
3.6974037
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36975780
3.693*752 J4986 3.6977523
36934631' 4987'3.6978394
3.693551 1 J4988 3.6979264
3.6936390 J4989!3 6980135
3.6937269 1 4990 3 698100 5
|4991 r3.6981876
4992j3.6982746
4993 3.6983616
4994 3 6984485
4995 3.6985355
3.6938149
6939027
3.6 39906'
49443.6940785
4945 3.6941663
3.6942541
6943419
494813 6944207
4949 3.6945174
'4950 3.6946052
4996U 6986224
499f
4998
4999
5000
3.6987093
6987963
3.6988831
16Q89rOO{
ARITHMETICS.
227
^\o
5001
5002
5003
5004
5005
50213.70079U2
5022)37008762
5023J3.7009632
5024^.7010496
502513.7011361
iuuo
5007
5008
5009
5010
5013
5014
5015
5017
5018
Lotr.
3.6990569
6991437
3.6992305
3 6993173
3.6994041
3.6994908
6995776
6996643
3 6997540
3 6998377
50113.6999244
50123.7000111i
3.7000977
3.7001843
3.700270'.
501637003575
3.7004441
3 7005307
5019J37006172
5020370070S7
5026(3.7012225
502713.7013089
5028|37013953
50293.7014816
5030.37015680
50313.7016543
5032;3 7017406
5033 3.7018269
5034J37019132
5035'3.7019995
50363.7020837
5037(3.7021720
503&3 7022582
5039J3.7023444
5040J 3 7024305
50413 7025167
50423.7026028
50433.7026890
5044
5045
5046
5048
5049
.7027751
17028612
No. | Log.
J No. i Lou.
505153.7033774
505213.7034633
5053;37035498
50543.7036352
505513 7037212
50563.7038071
50573.7038930
50583.7039788
50593.7040647
50603.7041505
5061|3.7042263
5062|3.7043221
50633.7044079
5064 3.7044937
50653.7045794
i5101 37076553]
51023.7077405;
51033.7078256 1
51043.7079107!
5105 3 7079957!
iNtj.f Li>y. |
5lll3.m«yl5
5152:8.71 19759
5153 3.7120601
5154B7121444
51 55J3 7122287
50/1
5072
5073
5074
5075:
5066|3.7046653
3.7047509
50683.7048366
50693.7049223
3.7050080
.7050936
.7051792
.7052649
.7053505
.7054360
5076(3.
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507813.
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,7057782
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7059492
7060347
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50911
50923
5093 3
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7064*17:
7065471
7066325
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70 8031
7068884
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7070589
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50473 7030333
3.7029472! 50963 7072294
5097 3 7073146
5098;37073998j
5099J3 7074850!
5050^ 7032914 1 5100)37075702'
3.7031193!-:
37032054
51063.7080808
510737081659
510837082509J
5109,3.7083359
511037084209
511137Ub5059.
511237085908!
5113 37086758
511437087607J
51l5j370ii8456
alid'S 71*312!
5157 8.7123971
5158 37124813
5159.37125655
5160 3 7126497
5116!o.708y305
511737090154
5118 , 37091003 l
5119 ! 3 7091851
512037092700 '5170
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5162
5163
5164
5165
6166
5167|
5168
5169
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o20l 3v 100869
520237161703
5203 37162538
5204 3 7163373
5205 37164207
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5207 3 7165876
520a 37166710
5209 3.7167544
5210 37168377
3 /U,3o9\ 5£ll o71o92.il
3.7128880! 5212 3 7170044
'3 7129021 '5213 37170877
3 7129862 J52U 37171710
3 7130703.I5215 3.717254:
5121 3.7093548 5171 o.< io5745
5122J3 7094396 j5172 37136585
51233709524415173 37137425
5124'3,709609: 5174 37138264
512537096939 ;51 75 3 7139104
5126 3.7097786
5127(37098633
512837099480
5129,37100327
513037101174
513137102020
513237102866
5133 37103713
513437104559
51353 7105404
513657106259
513737107096
51383 7107941
513937108786
514037109631
_ . 13-1 544 (a* 16 3.71/3376
J37132385 J5217 37174208
3 7133225 J5218 37175041
37134065 15210 37175873
3 7134905 5220 3, 7176705
0^13./ 177537
|5222 37178369
5223 3 7179200
J5224 37180032
'5225 37180863
5176 3 7139943
'5177 37140782
5178 3 7141620
5179 37142459
5180 37143298
518137144136
5182 37144974
5183 3 7145812
5184 37144665
51853 7147488
5186 3./ 148325
:5187J37149162
J5188 37150000
5189137150837
5190
5i41371l0476j:5l9l
5142 3 7111321 5192
51433 7112165J15193
51443,7113010^5194
5145 37113854 : 5195
51463 71 1469* ; 5196
51473 7115542 5197
5148 3 7116385 519P
51493 7117229"5199
51503 7118072i'520C J
37151674
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3 715334;
37154183
37155019
37155856
o./ 136691
37157527
37158363
3 715919^
37160033
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5227,37182525
6228 37183356
5229 3 7184186
=5230 3 7185017
5231 3 i 165647
15232 37186677
,'5^33 37187507
i5234 37188337
J52353 7189167
5236 3.7i§yyy6
5237J 37190826
5238.37191655
5239!37192444
5240 3 7193313
52416.4194142
i5242"a7194970
S243j37195799
5244 3 7196627
5245J37197455
■5246
5247
5248
V5249
5250
o.i \\ fc:C83
37199111
3 7199938
3 7200766
37201593
12 3
LOGARITHMICK
No-I Loer.
52513.7202420
5252*37203247
5253*3 7204074
5254 3.7204901)
52553.7205727 )
5256 3.7206554.
5257J3.7207380
5258 1 3-72082061
5259 3.7209032;
5260 3.7209S5 7J
526l'»3. 7210683
52623.7211508
5263 3.7212394
5264 3.721U59
5265,3 7213984
5206,3.7214809
5267|3.7215633
5268 3.7216453
5269:3.7217282
5270 3.7218106
No.| Log.
5301 3.7243578 15351]
5302 3.7244397||535:
5303,3.7245216:5353!
5304 3.7246035j 5354
5305 3. 7246854 J 5355
5306 3.7247672 15356 !
53073 7248491 5357J
53083.7249309)5358
5309 3.7250127 ! 5359'
53103.7250945 5360
5271,3.7218930
527237219754
5273 3.7220578
5274 l 3.7221401
5275 3.7222225
53113.7251763
5312 3.7252581
5313 3.7253398
5314 3.7254216
53153 7255033
3.7284350
3 7285161
3.7285972
3.7286784
7287595
3 7288406 5406
5361:
5362
5363
5364
53651
Los-
3 7289216
3.7290027
3.7290838
No.
5401
5402
5403
5404
5405
3.7324743
3.7325546
3.7326350
3.7327155
3.7327957
5407
5408
5409
3.7291648.5410
3.72924585411
3.7293268 5412
3.7294078 5413
3.7294888 5414
3 7295697i5415
5316 3.7255850
5317 3.7256667
53183.7257483
5319 3.7258300
5320 3.7259116
5321 3.7259933 5371
5366
53671
5368'
53691
.5370,
LO£.
3.7328760
3.7329564
3.7330367
3.7331170
3.7331975
3.7332775
3.7333578
37334380
3.7335183
3 7^35985
No. Log.
5451
5452
5453
5454
5455
3.7296507J5416 3.7336787
,3.7297316 5417 3.7337588
3.72981255418 3.7333390
372989345419 3.7339192
37299743 5420 3.7339993
5276 3.7223048
527713.7223871
5278 3.7224694
5279 3.7225517
52803.7226339]
5322 3 7260749
5323 3 7261565
532413.7262380
5325J3. 7263196
5326
5327
5328
53293.72.66457
5330
528113.7221162
52823.72279841
52833.7228806
52843.7229628
5285 3.7230450
5286'3.7231272i
5287
5283
5289
5290
*5372
5373
5374]
5375
3.7264012
3.7264827
3.7265642
3.7267272
5331
5332
5333
5334
5335
3.7232093
3.7232914',
3.7233736
37234557
5336
5337
5338
5339
5340
529113.7235378
5292 3.7236198
5293 3.7237019,
5294 3.7237839"
5295 3.7238660
5296
5297
>5298
5299
3.7239480
37240300
3.7241120
,3.7241939
S.7300552',5421 3.7340794
5376
5377;
5378
5379
5380
5381
'5382
!5383
3.7268087
3.7268901
3.7269916
3.7270530 5384
3.7271344 5385
"»' 5386
5341
5342
5343
5344
5345
.7272158
3.7272972
.7273786
3.7274599
3.72 75413
3.7276226
3.7277039
3.7277852
3.7278664
3 7279477
3.7364762
3.7365558
3.7366355
3,7367151
3.7367948
5456
5457
5458
5459
5460
5461
5462
5463
3.7368744
3.7369540
3.7370355
3.7371152
3.7371926
3.7372722
3.7373517
13.7374312
5464 C /375107
5465(3.7375902
3.73013605422
3.7302168 5423
3.7302977,5424
3.73037855425
3.7341595
3.7342396
3.7343197
3.7343997
3.7304593 5426
3.73054005427
3.7306208J5428
3.7307015 5429
3.7307823
5300 '37242759
5346
5347
5348
5349
5350
3.7308630
3.7309437
3.7310244
37311051
3 7311857
3.7312663
37313470
3.7314276
3.7315082
; 3.73 15888
3.7280290
3.7281102
3.7281914
3.7282726
3.7283538
5387
5388
5389'
5390
539l!i3.7316693
5392] 3.7317499
5393:3.7318304
5394 3.7319109
5395 3.7319914
5430
5431
5432
5433
5434
5435
3.7344798
3.7345598
3.7346398
3.7347198
3.7347998!
5466 3.7376696
546737377491
5468 3.7378285
5469 3.7379079
547037379873
5471
5472
5473
5474
5475
5477
5479
5480
3.7348798:
3.7349598
37350397
37351196 1
3.7351995
5436
5437
5438
5439
5440
5441
5442
5443
5444
5396
5397
5398
5399
5400
3.7320719
3.7321524
3.7322329 5448
3.7323133
3.7323938
5481
5482
5483
3.7380667
37381861
3.7382254
3.7383048
3. 7383841
5476 3.7384634
3.7385427
54783.7386220
3.7S87013
3,7387806
3 7352794
3.7353593
3.7354392
37355191
3.7355989
3.7356787
37S57585
37358383
3.7359181
5445 3.7359979
5446
5447
5449
5450
3.7360776
3.7361574
3.7362371
3.7363168
3.7363965
3.7388598
3.7389390
3.7390182
5484 3.7390974
5485(3 7391766
5486(3.7392558
5487 3.7393350
5488 3.7394141
5489 3.7394932
5490 3.7395723
5491
5492
5493
5494
5495
5496
3.7396514
3.7397305
3.7398096
3.7398887
3.7399677
3.7400467
5497 3.7401257
15498 3.7402047
5499 3.7402837
'5500'3.7403637
ARITHMETICS
229
550613.7408302, ^37447622
55073.7409151
55083.7409939
55093.7410728
55103.7411516
5511 £7412304
551237413092
5513 3 7413880
55143.7414668
55153.7415455
55163.7416243
55173.7417030
55183.7417817!
55193.7418604 1
5520 3-741 9391
5521
Lor:.
5557.3.7448404
55583.74491 85
5559.3.744990?
55603.7450748
556l'3.745l529
5562 ! 37452310
55633.7453091
55643.7453871 |56143.7492724|
5565 3 7454 652 ; 561 5j 3,7493498
5566^7455432 J5616 .3.74^4271
5567J37456212
.74201771
55223.7420964J
55233.7421750
55243.7422537
55253.7423323
55263.7424109
552737424395
55283 7425680
552937426462
55303.7427251
5531J3.7428037
5532)3.7428822
55333,7429607
55343.7430392
55^537431176
5536|3.7431961
553713.7432745
5538.3.7433530
554OI3.7435098
55433.7437449
5539;.3.7434314 55893.7473341
No., Log.
£0, ^_
55013.7404416 555l'3.7443>12
55023.7405206 5^523.7444495
55033.7405995 55533.7445277
55043.7406784 5554 ! 3.7446059
550537407573 555537446841
No.
5601
5602
5603
Log:.
3.7482656
3.7483431
3.7484206
56043.7484885
560537485756j
56063.7486531'
5607I3.7487306 1 ,
560837488080 1
5609374888541
|5610
5611
15612
'56131
3.7489609 ,
3.7490403*
3.7491177!,
37491 950!
5568'37456992 '5618 3 7495817
5569,37457772 J5619 3.7496590
557037458552
5571,37459332
5572^7460111
5573 37460890
55743.7461670
5575 3.746244 9
55763.7463228
55773 7464006
55783.7464785
5621 3.749*lo6
5622 3.7498908
5623 37499681
5624 3.7500453
5625 3 750122 5
5b2b 3.7501997
5590 3 747411 815640
joy;
559:
13 3.7476448 5643
5544137438232 5594'3.7477225 J5644
5547|S7440582 [5597 37479553 5647 37518178' '569? >.
554813 7441365 (5598 37480329 '5648 3751 8947* '5698 3.
5549 37442147 J5599 37481105 5649 37519716!J569C 3
K55Q13 74429301)5600 3.7481880 5650 37520484' 57Q(
56173.7495044
5620 3.7497363
{5639
5541|3>435881 15591 3 7474895 J5641 3.7513561 5691 37551886
5542 3.7436665 5592 37475672 ;5642 37514331, 5692 3.7552649
5545 3.7439016 1 3595 3 7478001 ;5645 3.7516639 569513 7554937 ! 5745
55463.7439799 J5596 3.7478777 '5646 O.7517409 569t 77555700 !5746
37556462 [5747
37557224 '57A&
37557987 ;574S
37558749 1 '5750
1 Log, j
0,7521253'
3.7522022
37522791
3 7523558
3 7524326
37325094
37525862
37526629
37527397
3.7528164
37529699
3.7530466
37531232
3.7531999
37532766
37533532
3.7534298
; ; 7535065
3.7535831
37536596
37537362
. _ 37538128
5674 3.753#893
3.7539659
po/oi 7540424
5677 3.7541189
5678 3.7541954
'56793.7542719
15680 3.754348 3
J5681 3.7544248
5721
5722
5723
5724
5725
5627 3.7502769
562837503541
5579 3.7465564 15629 3.7504312
5580 37466342 5630 3.7505084
5581*3.7467120 5631 3.7505855
5582 3.7467898 5632 3.7506626
5583 3.7468676 563
5584 3.7469454 5634 37408168 ,f 5684|3 7546541
5585 3.7470232 |5635 3 750893 9^685 3-7547305
5586 57471009 '5B36 3.7509710 0086 J.7 o4aoo9
5587 3.7471787 5637 37510480 15687 3.7548832
5588 3 7472564 5638 37511251: 5688 3 7549596
^5682 37545012
3 7507398|'5683 3 7545777
37512021 J568<
3.7512791.5690 3.7551123
3.7515101 H5693
37515370 5694
3 7550359
3.7553412 5743
3 7554175 |5744
No.
5701
5702
5703
5704
5705
Lop\
3.7559510
37560272
3 7561034
37561795
37562 556
3 7563318
3.7564079
5709
15711
! 5712
J5713
'5714
;5715
5/0(5
5707
570837564840
37565600
57103.7566361
3.7567122
3./*67882
3.7558642
37569402
7570162
;5716jj.f 570922
.571737571683
'5718:3.7572442
571937573201
;5720|3 2573960
5726
572
5728
'5729
o., 5/4719
37575479
17576237
37576996
. 757 7755
J.7573513
5732
5733
'5734
37579272
3.7580030
3.7580788
573013.7581546
'5731
o./ 5*2304
37583062
37583819
37584577
573337585334
j/J6
, : 5737
J5738
<5739
5/40
^/"3boUyl
3758684f
3.7587605
3.7588362
58911?
i574i
5 l 3&yti75
J742 37590632
37591388
37592144
3.7592900
3.7593650
37594412
3.7594168
1.7595923
3.759 667?
30
I
2St
L0GAR1THMICK
575137597434! 58U1
5752 3 7598189 ;5802 3 76C>5777\<5852 3 7673043
57611-3.76049/9
5762*7605733
5763
3 7606486 5813 3.7644003
57643.7607240
57653.760799:
5766
5767
5768
5769
3.7608746
37609500
3.7610253
3.7611005
5770 3.7611758
3.7612511
5771
5772
5773
3.7613265
3.7614016/5823
57743.7614768
577537615520
57763.7616272
57773.7617024
^778 37617775
577937618527
578037619288
5781
5782
5783
5784
5785
5786
5787
5789
5790
5791
5792
5793
5794
5795
5796
5797
5798
5799
5800
5814
'5815
37620030
37620781
37621532
37622283
37623034
37623784
37624535
578837625285
37626055
7626786
5804
5805
5753 3 7598044!j580:
5754!37599699r
5755:37600453
57563.7601208
57573760196$
57583 7603714
575913 7603471
5760'3 7604225
No, Lutf. ||No[ Lo?. ||i\o.| L,ug.
37635U29|(5851 37672301
3.7636526 J5853 37673785
3 7637274 f 5854 37674527'
3 7638022 15855 3 7675W9
5807
5808
5809
5810
5811
5812
5901
5902
5903
5904
58(58 £7638? /7poo6 376/6011,
-' 3.V 6395181 5857,37676752
37640266
37641014
37641761
3.7642509
37643256
3.7644750
37645497
5816
5817
5818
5819
15820
37646244
37646991
37647737
37648484
3 7649230
,5821 37649976
;5822(37650722
3.7651468
582437652214
58253765295S
582637653705
1582737654450
58283.7655195
'582937655941
583Q 3 765668 6
5H3i : 37657430
5832.37658175
583S37658920
{583437659664
5835 ! 3 7660409
I5d3g 3 766115
,5837.37661897
583837662641
583937663385
584037664128
37627536 5841J3.7664872
37628280 58423766561f
1.7629035 58433 7666351
37629785} 584437667102
3 7630534; 5845 37667845
584(>37668588
37631284 !
3.6632033
37632782
37633531
3.7634280
5847,3.7669331
5848,37670074
5849.37670816
5850 ! 3 7671S59
5858'37677494]
585937678235^
5860
376789761
566i 3767971/1
5862
5863
5864
5865
37680458J
37681199
37681940
37682680
5866
5867
5868
5869
37683421
3.7684161
37684901
37685641
5870 37686381
5871
5872
5873
5874
5875
Log. [liNo. ^Lug.
37709256: 5951 3.7745899 \
37709992! (5952 37746629
37710728: 5953 3.7747359
37711463; 5954 37748088
5905 , 37712199 , !5955
5906 ! 3 7712934, joy56
5957
5958
5959
59073.7713670 1
5908 , 37714405 l
5909;3.7715l40i
5910 3 7715875,
5911
5912
5913
377166101
o.774^547
37750276
37751005
37751734
5960 37752463
37717344
37718079;
5914J37718813
J5915 3.7719547
1591637720282
[591737721016
5918 , 37721750
59193.7 722483
5920 i 3.7723218
376871211
37687860
37688600
37689339!
37690079!
5tt/ 6 3769081 81
58773.7691557
587837692296!
587937693035 ;
588037693773'
5881
5882
37694512
37695250
588513769598a
5884)37696727
5885137697465
58*5(^37698203
5887(37698940
5888i3.769667fr
5889J3760O415
58903 7701153 '
589l'377018<*0i
5892137702627
58933.7703364
58943.7704101j
5895377048381
589637705575 1
589737706311
58983 7707048
589937707784;
59003.7708520!
5927
5928
5929
5930
5931
5932
5933
592137723951
7724684
37725417
5924 3.7726150
5925 37726183
5922
5923
592613.7727610
37728349
37729082
37729814
37730547
37731279'
37732011
37432743
593437733475
5935 b 7734207 ]
593613 7734939;
5937J3 773567
593937737133;
594037737864
5941'37738595
59423.7739326
594337740057
594437740?88
59453.7741519
5973
5974
37748818
5961
596$
5963
5964
5965
5966
5967
5968
5969
5970
5971 37760471
59723 7761198
3.7761925
37762653
37753191
3 7753920
37764684
3.7755379
3 7756104
3.7756832
37757560
37758288
37759016
37759743
59753 7763379*
5976
5977
5978
3776410^
3.7764833
3J7G5559
597937766285
5980 i 3.7 767012
5^1:3.7767738
5982J37768464
5983:3.776919(1
5984'37769916
5985J 377706 41
59863.7771367
5987J37772093
5938 37736402) 5988,37772818
598937773543
5990' 3777426 8
5991,3.7774993
5992!37775718
5993!3777644S
59943.7777167
599537777892
594637742249
5947 37742979
594837743710
94937744440
5950 3774517Cfl6000!3 7781512
59963.7778616
5997137779340
599837780065
599937780781
2l
ARITHMETICS
331
6001
6002
6003
6004
6005
6006
6007
6008
6009
6010
6011
6012
601
6014
6015
0016137793078
601713 7793800
60183.7794522
60193.7795243
6020 ! 37795965
jNo.
Log.
i.7782236
3.7782960
3.7783683
3. 7734407
37 785130
3-7785853
3.7786576
3.7787299
3.7788022
3.778874?
6106
3.7822576 6107 37838279
37823293 6108 , 37858990
3.7834009 6 ! 09137859701
60603.7824726 I 6110!3.7860412
37789467
3.7790190
37790912J
' 7791634 1 ;
3.7792356 f
6061 3.7825443| 011113.7861 123
6062 3.7826159p6il2l3.7861833
6063 3.7826876 S 11313.7862544
6064 3.7827592 '61 14 3 7863254
6O65I3 7828308 1611 513.7863965
6066
6067
6068
6069
6070
OJ^lj3.7796686
6022 3.7797407,
6023 3.7798129
60243.7798850
60253.7799570
'60263.7800291
6027;3.7801012
6028'37801732
602913.7802453
6030'3.7803173
60313.7803893
60323.7804613
5033*3.7805333
60343.7806053
6035 3.7806773
60363.7807492
6037J3.7808212
60383.7808931
60393.7809670
6040'37810369
60413.7811088
60423.78118Q7.
6043i3.7812526'
6044 , 37813245
6045 3.7813963
60463.7814631'
6047,3.7815400
60483 7816118
604937816836
60503.7817554
No.
0051
6052
6053
6054
6055
3.7818271
3 7818989
37819707
3.7820424
3.7821141
6056
6057
6058
6059
3 7821859
Lt>£.
£
v
No.j Log.
61013.7854010
161023.7854722
6103 3.7855434
61043.7856145
6105
3.7856857
378o756b
No.] Ley.
[No. 1 Lo£. t
16151 3 7889457
|6152|3 7890163
161533.7890869
6154;37891575
6155'S.7892281
37829024
3.7829740,
3.7830459'
37831171
3 7831887!
1
16156,3 7892986
6157 3 7893691
615837894397
6159:3.7895102
616037895807
^161,0.7696512
61623 7897217
,6163'3.7897922
|6164;3.7898628
61653.7899331
16201 $.,924617.
.62023.7925318
'6203 3.7926018
162043.7926718
£205 3 7927418
6206 3.7 9-^8118
62073.7928817
'62083.7929517
62093.7930217
62103.7930966
6071
6072
6073
6074
6075
6076
0116
6117
6118
6119
„„* 37866804
6120 3 7867514
3.7832602
S7833318
3.7834033
3.7834748
.7835463
o.? 004075
3.7865384
:61 663.7 900035
16167
3.7866095 ;6168 ! 3.7901444
■6J69 3.7902148
6170,3.7902852
6121 f 3.7808224
6122J3 7868933
6123 3.7869643
6124'3.7870352
6125(3.7871061
1.7836178
60773.7836892
60783.7837607
60793.7838322
6080 37839036 (6130J37874605
6081
37839750
6171137903555
6172J37904259
6173 ! 37904963
6174 , 37905666
6175;37§06370
6126137871770
6127 3.7872479
612837873188
6129137873896
6131:3.7875313
608237840464 6132'37876021
6083 3 7841178 f 6133;3.7876730
6084137841892 61343 7877438
6085 37842607 16135J3 7878846
608637843319
6087}37844034
6088'37844746
6089J3 7845460
60901 37846173
6691|37846886
609237847599
60933.7848312
609437849024
60953 784973^
613637878853
6137J3.7879561
613837880269
613937880976
61403 7881684
6141 37882391
614237883098
61433 7883805
6096
6097
6099
61913.7917608
6198*37918309
6193 3 7919011
61443 7884512^619437919712
37850450
37851162
609837851874
37852586
1610037853298
6145 3788521^
614637885926
6147;37886632
614837887339
61493.7888045
6150 ; 37888751
0176J3.7907073
161773.7907776
61783.7908479
6179!3J'909182
6180J37909885
618137910587
61823 7911290
618637911993
618437912695
6185 37913397
618637915099
61873 7914801
61883.7915503
618937916205
6190 3 7916906
6195 37920413
6196o.-9211l4
619713.7921816
6198|'j.792251f
519913 792321F
620013.7923917
0^113.7931615
6212 37932314
6213 37933013
621437933712
6215 3.7934411
62l6o./y35*l0
6217J3 7935809
6218 , 3.7936507
6219:3.7937206
6220'3 7937904
0221 37*38602
622237939300
6223'3.7939998
J 6224 ! 37940696
'6225:37941394
6226*37941091
6227*37942789
»6228 ! 37943486
I6229J37944183
i6230!37944880
;6231
'6232
|6233
6234
6235
37945577
37946274
37946971
37947668
37948365
0236j3.7i/49061
6237 3 7949757
|62383 7950454
763393.7951150
6240!3,7P51«46l
,6241(3.7^52542
6242'37953238
! 6243;3 7953933
6244 l 37954629
! 6245 l 3 7955324
6246 3./ 956020
6247*37956715
624837957410
624937958105
6250 3 7958800
232
LOGARITHMICK
No.l Lop:.
No. | Lot:
6252
6253
62543 7961579
6255
6256|3
62573
6258k
6259!3
62503
626113
6262!3
6263'3
62643
6265'3
62*63
62673
62683
6269*3
6270'3
3 7960190
3 7960684
\'o.( Lor.
|No. Log.
3 7962273
7y629b,
7963662
7964356
7965040
7965743
79o<5437
7967131
7967824
66'jl.i /u9WJ5-
6302 ; 3 799478416352
63033 7995473 16353 3
63043 799616211635413
;63053 7996rt51; 1 6.iJ5
163513 80284^1 1
8029105
802978'J ; ,
8030472;
8031155>|6405
64U13 8062478
6402 ! 3 806315?
640£
6306,3
5307,3
6som
J6309J3
631 ^3
3 8063835
6404 3 8064513
8065191
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7998288' 6357 3
79989171:6358 3
7999605] .6359 3
8000 v ")4i 636013
,'6jiiu3
631213
631313
80318J9J
8032522.
8033205!
803:^888!
80J4571
6406
6407
6408
7968517|i63143
7 6 ( >2U , 6315|3
800096^|j63Gl
8001670 6362
8002358 Ji 6363
8003046j:63643
8:)03r.°>4! ! .6365!3
8035264
8035937
8036619
8037302
8037984'
6J713
62723
6273*3
6274'3
6275-3
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7970597
7971290
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7974753
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f 6316i3
6317|3
163183
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627713
62783
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7977521
7978213
7978905
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6281 ; 3
6282,3
62833
6284'3
6285 3
7980281
7980979
7681671
7982362
7983053
62863
628713
6288!3
62893
6290 3
7983744
7984434
7985125
7985816
7936506
6291 3
62923
62933
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62953
7»67 M
7987887
7988577
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7088957
62963 7990647
6397j3 7991337
62983 799202?
62993 7992716
6300'3 79934051
8U04421
8005100J
8005796
8006484
8007171
636o!3
63673
63683
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6370J3
6321
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16324
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8010605!
8038666,
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8040030'
8040712
8041394
63263 8011292
8011978
8012665*
8013351
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6328
6329
6330
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63343
6335 3
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6338
6339
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6341
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6344
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63463
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6348
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63503
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8015409
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8016781!
8017466
8018152
8018837!
8019522
8020208'
8020893
6371J3
63723
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63743
6375\3
8042076|
8042758:
8043439
8044121
8044802
6381
6382
6383
6384J3
6385
63863
63873
63883
63893
63903
8021576!
80£2262 :
80229471
8023032!
6391
63923
6393
6394
3 8024316-6395
8025001
8025685
8026369
8027053
8027737'
6396 3
6397 3
63983
6399 3
6400 3
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3 80658o9
3 8066547
3 8067225
6409:3 8067903
641013 8068580
6411J3
64123
641313
641 4'3
6415 3
64163
6417 3
64183
641913
6420*5
8069258
8069935
8070613
8071290
8071967
6451
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6461}3
6462 ! 3
6463 ! 3
6464!3
646513
8072644
8073320
8073997
8074674
8075350!
64213
642213
64233
64243
64253
8076027)
9076703-
8077379
8078055
8078731
6376'3 8045483] 6426'3 8079407
63773 8046164 64273 8080083
63783 8046845164283 8080759
63793 8047526 rt «4293 8081434
63803 8048207 6430}3 8082110
8048887
8049568
8050248
8050929
8051609
8052289
8052969
8053649
8054329
8055009
8055688
8056368
8057047
8357726
8058405,
6431|3
64323
6433*3
64343
643513
8082785'
8083460
8084135
8084811
8085485
643613
643713
6438'3
6439.3
6440 ! 3
8086160;
8086835 1
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8088184
8059085]
80597631
8060442
8061121
8061800
644113
6442-3
6443J3
644413
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644713
64483
6149,3
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8089533;
8090207
80908811
8091555:
8092229|
6496 3
8092903,'
8093350
8094250
8094924;
S095597'65003
3 8096270
3 8096944
3 8097617
6454'3 8098290
6455»3 8098962
64563 8099635
6457 ! 3 8000318
64583 8100980
6459'S 8101653
6460*3 8102325
8102997 i
8103669
8104342
8105013
8105685
6*6613
6467|3
64683
64693
64703
8106357
8107029
81077tX)
8108371
8109043
647i;3
6472J3
647313
6474'3
6475 3
6476
3 8113068
8113739
64783 8114409
3 8115080
64803 8115750
64773
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64853
64863
6487.
64883
6489
64903
6497 3
64983
6499
8109714
8110385
8111056
8111727
8112398
8116420
8117090
8117760
8118430
8119100
1*119759
8120439
8121108
8121778
8132447
64913 8123116
64923 8123785
6493 3 8124454
6494 3 8125123
6495 3 8125792
8126460
8127129
8127797
8128465
8129134,
ARITHMETICS.
23*
6501
65023.
3.8129802!
8130470!
650313*8131137
65043.8131805
6505 3.8132473
6506
6507
3.8133141
3.8133808
6508T3.8134475
6509|3.8135143
651013.8135810
65113.81^0477
65123,8137144
6513 3.8137811
65143.8138477
65153.8139144.
No
Log. (
6556 3.8166389
3.8167052
3.8167714
3.8168376
65603.8169038
6557
6558
6559
6501
6562
6563
651$3.8l39dll|
6517;3.8140478
65183.8141144
6519-3.8141810
65203.8142476
6521^3.814^14^
65223.8143808,
6523;3.8144474 i
65243.8145139i
65253.814580?
65263 8146471
6527|3.8147136
6528^.8147801
652913.8148467
6530;3 8149132
65313 8149797
65323.8150462
6533 3.8151127
65343.8151791
65353.8152456
6536
6537
6538
6539
6540
3.815312U)
38153785
3.8154449
38155113
'138155777
6541
6542
6543
6544
654b
6547
6548
3.8156441
3.8157105
3.8157769
3.8158433
6545 3.8159096
3.8159760
38160423
38161087
6549^3 8161750
6550 3
No.
6551
6552
6553
65543.
6555
Lo£.
3.8163076
3.8163739
3.8164402
1.8165064
3.8165727
3.8169700
3.8170362
3.8171024
65643.8171686
6565J3.8172347
6566-3 8173009
6567)3.8173670
6563-3.8174331
6569|3.8174993
657038175654
6571^38176315
6572J3.8176975
6573!>8177636
6574:3 8178297
6575.3 5178958
No.
6601 3.8196097
6602 3.8196775
6603 3.8197413
6604 3.8198071
6605 3.8198728
6606
6607
3.8199380
3.8200043
66083.8200700
66093.8201357
66103.8202015
6611
6612
6613
6576J3 81796181
657*3.8180278!
657a : 3.8180939!
6579|3.8181599
65803.8182259!
6581 3.8182919J
65823.81835791
6583 3.81842391
65843-8184898
6585J3.8185558
5861;
658613.8186217
6587i3.8136877
6588!3.8187536
65893.818S195
65903.8188354
659a 38189513;
659VJ3.8190172I
5593 3.8190831
65943.8191489
6595 3.8192148
6596,3.8192806
6597 3*8193465
6598.3.8194123
6599 ^.8194781
8162413W6600 3. 8195439
Lo*.
3.3202672
3.8203328
3.8203985
66143.8204642
66153.8205298
66163.8205955
66173.8206611
66183.8207268
6619-3.8207924
6620'3.8208530
6621 3.820 l J236
6622 38209892
6623 3.6210548
3.8211203
38211859
6624
6625
6626
6627
6628
6629
6630
6631
663!
663i
663'
6635
6637:
6638
6639
6640
6642
6643
6644
6645
6646
6647
6648
6649
6650
3.8212514
3.8213170
3.8213825
3.8214480
38215135
3.8215790
6632|a8216445
.8217100
6634)3.8217755
3.8218409
66363.8219064
3.8219718
5 8220372
58221027
5.8221681
6641 3.8222335
3.8222989
3.8223642
3 8224296
3.8224950
3.8225603
.8226257
3.8226910
3.8227563
3.8228216
I No. * Log. |
6651 13,
66523.
6653'3.
6654J3-
6655\3.
,8228869 i
8229522
8230175!
8230828'
8231481'
670113.8261396
6702J3.8262044
6703|3.8262692
6704*3.82633401
6705'38263988
6656 ! 3
665713.
66583,
66593.
6660 3.
8232133!
,8232786.
,8233438
8234090 :
3234742'
66613.8235394
6662 3 8236046
666338236698
66643.8237350
66653 8238002
66663.8238653
66673.8239305
66683.8239956
66693.8240607
66703.8241253
6671 3 8241% -J
'6672 : 3.8242560
'66743.8243211
6674*3.8243862
6675 3 8244513
6676 3.8245163
,66773.8245814
66783 3246464
16679 3.8247114
; 66S0 3.8247765
6681 3.8248415
66823 8249065
66833.824971
66843.8250364'
6685 3.8251014:
668638251664'
6687 3.825231
66883.8252963
66893.8253612;
66903.8254261 s
66913.8254910
6692i3.8255559 t
6693
6694
6695
6696
6697
6698
6699
6700
3.8256208J
3 8256857!
3 8257506
3.&258154!
3.3258803!
3.8259451i
3.82601001
3 8260748'
No*| Lor.
6706 j 0264635
6707,3 3265283
6708 3 8265931
6709J3.8266578
671013.8267225
6711
6712
6713
6714
3.8267873
3.8268519
3.8269166
3.8269813
6715 38270460
6ri6i3.827ll07
6717|3.8271753
6718 ! 3.8272400
6719)3 8273046
6r203.t>~"3693
67 2113.82? 4339
6722
6723
£8274985
3.8275631
672413.8276277
6725*3 8276923
6726J3 8277569
67273.8278214
3.8278860
3.8279505
6728
6729
6730 3.8280951
6731
6732
6733
6734
6735
,8280769
3.8281441
8282086
3.8282731
38283736
6736J3.8284021
6737|3.8284665
6738J3.8285310
673913 8285955
6740'[3 8286599
67411J.6237243
6742;3 8287887
6743 ( 3.8288532
6744i3.8289176
6745 J 3 K2898 20
6746 o b^ii0463
674713 8291107
6748 ! 3 8291751
6749'3.8292394
6750 3 8203830
334
LOGARITHMIC*
6751 3 82*i(x>i j68Ul
67523.8294324 6802
67533 4294967,6803
6754 3.8295611 I |6804
6755 3J296254
67oo 3 8jy6oyo
3.8297539
3.8298182
3S298824
6760!3 8299467
676l!3.83lK>iu*
676213.4300752
6763J3.8301394
67643*8302036
6765!3.8302678
No.f Loe:. .No.
3.8325728
3.8326366
38327005
3.8327643
69053.8328281
6807
6808
6809
6813
68153 8334659)6865
670oj3 8303320
6767 3-8303962
67683.8304604
67693.3305245
6770' 3.83 5887
6771|3.83J6o^o
67723.8307169
67733.8307811
6774 , 3-8308452
6775(3.8309093
68l0j3.8335296
6817|3.8335933
68183.8336570
681938337207
6820 ! 3.8337844
67763.8309/^4
67773.8310375
67783.8311016
67793.8311656
67803.8312297
6781]3.83129o7
6782 ! 3.8313578
6783'3.8S14218
67843.8314858
6785 3.8315499
I.o^.
6o\>0 3.8328yiy
3 832*558
3 8330195
3.8330833
6810 3 8331471
6811 3.833^oy
2.8333384
68143 8334021
No., Love-
68 J 1,3.8357540
6852:3^358174'
6853 3.8358607J
68543.8359441 1
6855 3.8360075'
69013.838 120
69023.8389750
69033.8390379.
69043.8391008
6905 3.8391637
68o6 3.8360708
6857|3 8361341!
6858 3 8361975i
685913.8362608
6860'3835324l!
6861
3.8^63874
6812 3 8532746*6862 3 8364507!
16863 3.8365140'
6864
38365773:
3 8366405
6866 3.8367038
6867
'6868
6869
8367670.
3.8368303
3.8368935^
6870 38369567;
6821 , 3.83384riU
68223 8339117
68233.8339754
68243 8340390
6825 3.8341027
68263.8341663
68273.8342299
68283.8842935
68293.8343571
68303.8344207
6871
16872
J6873
6874
16875
3.8370199;
3.8370832;
3.8371463 1
3.8372095 6924'3.840354ll 6974
m?6
6877
6879
68313 8344443
6832 3.8345479
683338346114
6834*3.8346750
6835,38347385
6786 3.831613* 16836,3.8348021
6787 3.8316778 i6837;3.8 348656
6788 3 8317418 I683&3.8349291
6789 3 8318058 1683938349926
6790 3.8318698 168403.8350561
6/yi 3.83iyoo/i
67923.8319977
67933 8320616
67943 8321255
67953.3321895
6841'3.8351iyo
6842:38351831
6843'3 8352465
6844 ] 3.8353100
68453.8353735
67963.832268-*
679738323173
67983.8323812
67993 8324450
6800 3.8325089
6882
6883
6885
6886
6887
6888
69li 3.8395409.
69123.83960371
69133.8396666!
69143.8397294'
6915 3 8397922
3.8372727,
3373359;
3.8373990
68783.8378622'
3.8375253:
68803.8375884
6881 3.8376510
3.8377147
3.8377778
68843.8378409
3.8379039
3.8379670
3.8380301
3.8380931
6889 3-8381562
6890
3-8382192
6891
,6892
.8382822
3 8383453
68933.8384083
6894
6895
68463.835436y
684713 8355003
6848'3 8355638
684913.8356272
6850 , 3.8356906
No. f Logi
No.
6951
6952
6953
6954
6955
69063 83*2266
69073.8392895
6908*3.839352^
6909 3.8394152'
6910 3.8394780'
69o6
6957i
6958
6959
6960
6961
69621
6963 1
6964
6965!
6916 3.8398550
6917S.8399178
69183.8399806
6919:3.8400453
69203.8401061'
692i;3.8401688
69223.8402316'
6923 8.8402943 1
6925'3. 8404198 »
6y2b;3~8404825
6927
6928
2929
6930
3.8405452
38406079,
3 8406706 1
3 840733 2]
3.840795yi
,6933
! 6934
6935
6931
169323.8408586
3.8409212
3.8410988
3.8410465
3842047,
3 £421098
3.8421722
38422347
3.8422971
3.8423596
38424220
3.8424844
3.8425468
3 8426092
6975
6976
6977
6978
6979
6980
Log.
3.8426716
38427340
3.8427964
3.8433588
3.8429 211
38429835
3.8430458
3.8431081
3.8431705
38432328
6981
6932
6983
69841
6985
3.8432951
3.843S574
3.8434197
3.8434819
3843544 2
3.S436061
.8436687
3.8437310
3.8437932
3.8438554
3.8439176
3.8439703
3.8*10420
3.8441042
3.844166>
6936 3.8411091jj6986i
16937 3.8411717l|6987i
6938 3.841234316988
6939 3 8412969! 69891
6940;33413595j!6990'
3.8384713
3 8385343 ;
8385973;
3.8S86602
3 838 232,
38387861
6896
6897
6898
6899
69003.8388491
694113 8414220J6991,
6942J3 84t4846p6992
694^3 ?415472| ! ,6993
694438416097 6994
6945 3^416723 6995
3.8442286
38442907
3.8443529
38444150
3.844477 2
3.8445393
3.8446014
38446635
3 8447256
3.8447877
3.8448498
3.8449119
69463-8417348 6996
69473'8417973||6997
6948 3-841859816998 '3.8449739
6949 3-8419224 ,6999 i3.8450360
6950 S.a±19848! l 7000 t3.843Q98Q
ARITHMETICS
3»
No.f Log* ,
700l3lJ45llKiI
0023.8452221
70033.8452841
70043 8453461
70053.8454081
70063.8454701
70073.8455321
70083.8455941
70093.8456561,
701034457180
70113 8457800
70123.8458419
70133.8459038
70143.8459658!
7015|3.8460277
No.. Log. No.| Lour.
70513-8482507
70523.8483123
70533.8483739
70543.8484355
7055 ] 3 8484970
7056
7057
71013.851319^
71023.8513807
71033.8514418
71043.8515030
71053.8515641
3.3485586
3.848620r
70583 8486817
70593.8487432
70603.8488047,
70163.8460896
70173.8461515
70183.8462134
70193.8462752
70203 8463371
70213 8463990
70223.8464608
3.8465227
3.8465845
3.8466463
7023
7024
7025
7036(3.8467081
70273.8467700
70283.8468318
3.8468935
3 8469553
7029
7030
7033
7034
7041
7042
7043
7044
7045
7046
7047
70313.8470171
70323 8470789
38471406
3.8472024
7035*3.8472641
70363.8473258
7037 3.8473876
70383.8474493
7039 3.8475110
70403 8475727
3.8488662
3.8489277
3.8489892
70643.8490507
70653.8491122
7061
7062
7063
7067
7068
7071 3.8494808
7072
7073
7074
7075
6077
7078
7079
7080
7106-3.851625.
71073.8516863
71083.8517474
710938518085
71103.8518696
71113.8519307
71123.8519917
7113 3 8520528
71143.8521139'
7115 3 8521749
7066)3.8491736
3.8492351
3.8492965
70693.8493580
70703.8494194
71163.8522359
71173.852*970;
71183.8523580
71193.8524190!
712038524800
71213.
3.849542*71223.
71233.
71243.
71253.
3.8496037
3.8496651
3.8497264
7076)3.8497878
.8498492
r.8499106
3,8499719
8500333
7081l38500946
7082
7083
7085
3.8501559
3.8502172
7084,3.8502786
3.8503399
7086:3,8504011
7087^3.8504625
70883.8505237
7089'3.8505850
3.8476343J 70913.350707 5.
3 8476900 7092,3. 8507687;
3.8477577! 7093:3.8508300
3.8478193
3 8478810
709413.8508912
70953:8509524
No. 1 Lor.
7151,8.8543668
71523.8544275
i7153'3.8544882
715438545489
71553.8546096'
7156U8546703
8.8547310
3.8547917
3.8548524
3 8549130
7157
7158
7159
7160
8525410
,8526020
8526629)
8527239:
8527849,
71263.
71273
71283
712*3.
71303
,8528458
8529068!
8529677
.8530286
8530895
7161
7162
7163
7164
7165
3.8549737'
3.8550343
3.8550950
3 8551556
3.8552162-
7166
7168
3169
7170
7171
7172
7173
7174
7175
7176
7177
3.8552768;
71673.855;
374
3.8553980 1
3.8554586
3 85551921
3.85557971
3.8556403
3.8557008:
3.8557614
3 8558219
8.8558824;
3.8559429J
71S1 3.8531504
7132 $.8532113
71333.8532722
31343.8533331
713538533940
71363.8534548
7137J3 8535157
71383.8535765
7139|S.85S6374
7090j3.8#)S462 ]7140,3 8536982
7141
7142
7143
3 8479426 70963 8510136
3.8480043J 7097J3 8510748
7048*3.8480659 70983.8511560'
7049 3.8481275 7099J3.8511972
705013 8481891 1 7100,3.8512583
|71463
1714; -
(71 18)3
7149
3,8537590
3 8538198
3 8*38807:
714438539414
7145 3 8540022!
8540630
38541238.
8541845
3.8542453
8543060
7178.3.8560035
7179,38560640;
7180 3 8561244:
17181 3 85*1849
71823.8562454
7183 3.8563059
718438563663
7185 3.8564268
7186 l 3.8564872
718713.8565476
71883.8566081
7189J3.8566685
7190 3 8567289
7191 3 8567893
7192 3.856784?
7193'3.8569101
71943,8569704
7195,3.8570308
Nof L
£•
7201io.a573i,28
7202o.857453I
72U3 3.8575134
7204 3 8575737
7205'3.8576340
7206 3 8576943
7207*3.8577545
7208*3.8578148
7209 8S57875
7210 38579353
72113 8579955
7212,3-8580557
7213 3.8581156
7214 3.8581761
7215 3,8582363
7^16 3.8582965
7217i3.8583567:
7218 3.8584169
7219,3.8584770
72203,85&5372
7221 3.6585973
7222 3.3586575
7223 3.8587176
7224 3.8587777
7225 3.8588379
7226,3.6588980
7227|&8589581
7228,3 8590181
722938590782
7230 3 8591383
7231.3 8591984
723238592588
7233-38593185
7234 , 3.3593785
72353 8594385
7236;3.6a94986i
7237
7238
7239
7240
£8595986
3.8596186
3.8596786
3 8597386
7241 8 8597985
7242
7243
7244
7196 3.8570912
7197 18 8571515
7198;3.8572118
7199;3.857272
720013.85733^5
7245 3.8600384
7246
7247
7248
724b
7251
3.8598585
3 8599181
3.8599784
3.b'C00983
38601583
3 8602182
38602781
3.8603380
S96
LOGARITHMICK
No. ' Loq:. j)No.| Logr.
7251 3
7252-3.
72533.
7254:3
7255*3
7250 a
7257|3.
7258,3-
7259 3.
7260 3
860J9/y;|7301 3 8633823
8604578| 7302 3.8634418
.8605177173033.8635013
8605776 7304*3.8635608
8606374 1 7305 3.8636202
.8606973 17306 3.863*6/97
,8607571; 7807 3.8637391
•86081701,73083.8637985
.8608768! 7309 3.8638580
.860936617310 3.8639174
726l[3.860y9o4
72623.8610562!
7263 3.8611160;
72643.8611758:
7265 3 861235 6J
7206,3.8612954!
7267;8.8613552j
7268 3.8614149
7269,3.8614747
7270 3.8615344!
7311 3.8639768
73123.8640363
73133.8640956
7314 3.8641550
7315 3.8642143
73163.8642737
73173.8643331
7318 3.8643924
7319 3.8644517
7320 3.8645111
727113.8615941
7272 , 3.8616539
7273 3.8617136
( 7274;3.8617733
7275,3.8618330
7276 3.8618927
7277 3.8619524
7321 3.8645704
73223.8646297
7323 3.8646890
7324
7325
7278
7279
7280
3.8620121
3.8620717i
3.8621314
7281
7282
7286
7287
7283
7289
7290
3.8621910
3.8622507;
72833.8623103
•7284 3.8623699
72853.8624296
3.8624892
3 8665488
3.8626084
3.8626680
3 8627275
7291
7292
7293
7294
7295
7341
a8627871
38628467
3.8629063
3.8629658 7344
3.8630253 7345
3.864748:
3.8648076
7326
7327
7328
7329
7330
3.8648669
3.8649262
3.8649855
3.8650447
3.8651040
7331
7332
7333
73343,
7335
3.8651632
3 8652225
3.8652817
1.8653409
?.8654001
7337
7338
7339
7340
7343
7296,3.8630848
7297 3.8631443
729813.8632039
7299J3.8632634
7346
7347
7348
7349
^00-3.8633229 7350
No.
7351
7352
7353
73543.
7355
Lo*.
3.8663464
38664055
3.8664646
8665236
3.8665827
7356i3 866*417
73573.8667008
73583.8667598
7359J3.8668188
736013.8668778
74063,
7407
17408
,7409
,7410
7361J3.8669368
73623.8669958
7363J3.8670548
73643.8671138
7365;3.8671723
7366 3.8672317
73673.8672907
38673496
3.8674086
73703.8674675
7368
7369
7371(3.8675264
73721 3.8675853
73733.8676442
73743.8677031
7375 3.8677620
73763.8678209
7377j3.8678798
7378 ! 3.8679387
7379
7380
3.8679975
3.8680564
7381
7382
7383 3.
7384
7385
73363.8654593
3.8655185
3.8655777
3.8656369
38656961
3.8657552
7342 3 8658144
3.8658735
3.8659327
3 8659948
3.8660509
3.8661100
38661691
38662283
8662873
7386
7387
7388
7389
7390
3.8081152
3.8681740
•8682329
3.8682917
3.8683505
3.8684093
3.8684681
3.8685269
3.8685857
3-8686444
7391
7392
7393
3.8687032
3.8687620
3.8688207
No.
7401
7402
74033.
7404
7405 3,
7411
7412
7413
7414
7415
3.8698768
3.8699354
3.8699940
3.8700526
3 8701112
7416
7417
7418
7419
7420
7421
7422
7423
7424
7425
7426
7427
7428
7429
7430
7394;3 8688794
[395
7396
7397
;7398
17399
*7400
3.8689382
7431
7432
7433
7434
7435
Log.
.8692904
.8693491
.8694077
.8694664
.8695251
,8695837
18696423
.8697010
.8697596
.8698182
3.8701697
3.8702283
3.8702868
3.8703454
3.8704039
38704624
3.8705210
3.8705794
3.8706380
3.8706965
a8707549
3.8708134
3.8708719
3.8709304
3.8709888
3.8710423
3.8711057
3.8711621
3.8712226
3.8712810
7436
7437
3.871S394
3.8713978
7438 3.8714562
,8715146
3.8715729
r 439 3.
7440
7491
7492
7494
7495
3.86B9969
3.8690556
3.8691143,
3.8691730J
38692717
496
7497
7498
7499
7500
3.8745398
3.8745978
74933.8746557
3.8747137
3.8747716
No.
7501
7502
7503
7504
7505
3.8751192|
3.87517711
3.8752349^
3,8752928
3.8753507
7506
7507
7508
7509
7510
3.8754086
3.8754664
3.8755243
3.8755821
3.8756399
3.8756978
3.8757556
7513 3.8758134
75143.8758712
7511
7512
7515
7516
7517
3.8759868
38760445
7518(3.8761023
3.8761601
38762178
7519
7520
7521
7522
7523 3.8763911
7524 3,
7525
7526 3.8765642
7527
7528
7529
7530
3.8748296
3.8748875'
3.8749454
3.8750034
3.8750613
7531
7532 3.
7533
7534
7535 3.
7537
7538
7539
3.8759290
3.8762656
38763333
.8764488
3.8765065
3.8766219
8766796
(.8767373
3.8767950
3.8768526
i.8769103
3.8769680
3.8770256
,8770833
75362.8771409
3.8771985
3.8772561
3.8773137
7540 3.8773713
7541
7542
7543
75443
7545
7546
7547
7548
7549
3.8774289
3.8774865
3.8775441
8776077
3.8776592
3.8777168
3.8777743
3 8778319
3 8778894
75503 8779470
ARITHMETICS.
237
fe'*' ,- i^g. J >Jo.| Log. IjNo.y Log.
7651
7652
7653
'7654
7655
o.bbo/182
7551 3.87bU045ft7601 3.8808707
75523.8780620! 760213.8809279
7553 3.8781195 760313.8809850
7554 3 878l770l!7604;3.8810421
7556 3.8782345 ) 7605*3. .881 0992
75563\8782919! 7(3063.8811563 .# u56> oa*uul^
7557 3.8783494! 7607
7558 3.8784069:7608
75595.878464-317609
3.8837750
3.8838317
3.8839885
3,8839452
Nx>.
L<<g. ■
7701 3.'8S65471
770213.8866035
7703 3.8866599
7704'3.8867163
77053 8867V 26
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7565 3 8788089 :7615 3 8816699 : j 7665 13 88 45122 7715J 3.8S73359
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3.8846255 7717 3.8874485
3 8846821' 771 &3.8875048
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7570 3 87.90959 ! 7620 3.88195 50 ; |767() £8847954 7720 3.8876173
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SI
•2SS
LOGARITHMICK
No I L^fr.
7fi0li3.8929503j
780213.8922059
7803;'3.8926616j
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7819 3.8931512
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7824^3.8934288
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ARITIIMETICK
• 339
No. Loir.
8U51
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Lot;. | No.j Lov^.
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240
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92*4958 849
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8531
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8536
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9311017
9311526
9312035
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9314070
9314579
9315087
9315595
9816104
9316612
9317121
3 9317629
3 9318137
3 9318645
3 9319153
3 9319661
ARITHMETICS
**l
8551
8552
8553
No
8556
855/
855&
855'.
8560
L:>%. | No. Log |
393A>169i
£9320677]
3.9321185)
8554 3.9321692!
8555 3.9322200!
3.934546^'
3.9345994
3.9346499
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8605J3.934750'' 1
8601
8602
8603
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3.9322708!
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86J6!3.95480i;<
3.932373&
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9,3.
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8608
8609
393247381
86103 9350032
8oDli3.'J3i5,243!
896213.93257521
8563|3.9326259|
8564 ! 3.9326767!
8565 ft /?327274 !
85oO;3.9527/bi:
8567J3.9328288'
8568 3.9328795!
8569 3.93293011
85703.9329808'
8611
8612
861,
8614
8616
8617
8618
8619
8520
8571J39330315
8572(3.9330822
8573j39331328
8574 3.9331835
8575 3.9332341
8621
85763 9332&43
85773.9333354
3 9333860
3 9334367
3.9334873
8578
8579
8580
8581
8582
8583
8584
8585
.9336391
3.9336894:
3.9337403"
8586-3.9:i3/9U9,
8587 39338415
8588 3.9338920'
8589 3.9339426 1
8590 ! 3.9339932'
8591 3.93*0* j/;
8592 3 9340943:
85933.934 1448;
85943.93419531
859.5,39342456
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!8 197:3 9343469
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86523.9371165
8653 39371667
86543.9372169
8655 3.9372671
3.9348518
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86ib'3.93731/2
8657S.D373674
3558 f 3.93/4176
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3660 3 9375179.
3.9350050
3.935 1040
3 9351544
3.9352C4J;
86153.9352553
8661 3.9375680
3662 3.9376182
3663 3.9376683
8664c.')377184;
8665 3.9377686
3.935^05/
3 9353561
3.9354065
3.9354569
3 9355073
3 9355570
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3623 >*9356584
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8625 3 9357591
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8629
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3 9358U95
3.9358498'
3.9359101
3.9359605,
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3.93358851 3632!3.9361114;
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3635=3.9362623
3636 3.9363120
3637J3 9363629
36383.9364132
8639 3.9364635
86403.9365137
86a'3.9365640i
36423.9366143;
3643J3.9366645!
3644 ! 3.9367H8 !
86463.9368152
8647(3.9368655
86483.7369157
8649J3.9369659
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87023.9396191
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3.9420577
39421073
3.9427569
3.9422065
8755;3.9422562
86663 9378187
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86683 9379189
8669 3.9379690
8670 3. 938019 1
8671 3.9380692
86723.9381193
8573 39381693
86743.9382194
8675 3.9382695
3676 3.9383195
8677 3.9383699
86783 9384196
86793 9344697
86803 9385197
8081 3 9385698
3682 3.9386198
86833.9386698-
36843.9387198
8685 39387698
8686 3 9388798
36873 9388698
36883.9389198
86893.9389698
86903.9390198
8691 3.9390697
8692
8693
3694
8695
3.9391196
3.9391697
3.9392196
3.9392696
3696
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3.9393195
39393695
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37003.9395193
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8707 3 93G8685 87o7|3 9423553
8708 3 9399184! Hi 58 3.9424049
8709 3.9399683 : 8759 3.9424545
8710 3940018':! 8760 3 9425041
3.9425537
3.9426032
3.9426528
3 9427024
39427519
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8713!3 9401677|I8763
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8718 3.9404169
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87213 9405663
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8724 3.9407157!
8725 3.9408654
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877
8774
8775
3 9428015
39428510
3.9429005
3.9429501
3.9429996
> 9430491
5.9430986
1.9431481
S.9431976
) 9432471
8731
8732
873S
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8735
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9434945
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:.9411137j|8782,3.9435934
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8737
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,9413623: : 8787 3 9438406
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9414617
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8742*3.9416108
8743 3.9416605
8744 3.9417101
8745 3.9417598
874613 9418095
8747 39418591
3748 39419038
8749 3.9419584!
8750 3.94200811
8788 39438900
8789 3.9459394
8790 3 9439889
8791 3:9440383
87923.9440877
8793 3.9441371
87943.9441865
8795 "3 9442358
8796o.9442852
8797
8798
8799
8800 1
3 9443346
3.9443840
3.9444333
3.9444827'
S46
LOGARITIIMICK
No
8801
&802
8803
8804
8805
880o3
3807
8808
8809
8310
L*>£.
,No.
jOiX
;.9445J20,885lfc
.94458i4 ( 8852 3.
.9446307, 8853'3.^. . ^^ „.^„„^
.9446800 8854J3.9471395, 1 8304 3.9495852
.9447291- |8S55!3.9471886j 8905 3 9496339
94699231
9470414
9470905
.No., L<K>
89J1 3.9494388
8902 3.9494876
8903 3.9495364
No. |
SSI's"
89523,
89533,
8954'3.
8955 3
.9447; 37,88^13 9472376.^^06 3.9496827j
8857 39472866 8907 3.9497315;
.9448280
9448773
.9449266
9449759
88583
8859 ; 3.S
9473357!
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8860[3.9474337 :
88113.9450J52
681? 3.9450745
8813:9451238
8814(3 9451730
881513 9452223
8861 3.94? .382/.
8862 3.9475317,
8310.3.9452/ lo
88173.9453208
8818'3.9453701
8819 3.9454193
88203.9454686
^
8908 3.9497802
8909 3.9498290
8910 3 9498777,
8863
3.9475807
88643.9476297J
8865
39476787
8366
7867
8868
8869
8870
3.9477277i
J9477767;
39478357;
3.94787471
3.9479236
88213.9455176
882239455671
8823 3.9456163
882439456655
88253.9457147
8871
8872
8873
8874
8875
88263.9457639
88273.9458131!
8828 3.9458623
8829|3.9459115
88303.9459607,
8877
8878
8879
8880
89563.
8957t3.
8958'3.
8959J3,
8960 ! 3
8911 3.9499264
8912 3.9499752
8913 3.9500230
8914 3.9500726
8915 3.9501213
8916 3.9501701J
8917 3.9502188
8918 3.9502675;
8919 3.9503162
8920 3.9503649
3.9479726
3.9480215
3.9480705
3.9481194
3.9481684
89213.9504135
89223.9504622;
89233.9505109,
89243.0505596
892539506082;
88763.9482173
3.9482662
88783.9483151
.9483641
3.9484130
8831 3.94t>0uyy, 8881
8832 39460591:8882
8833.3.9461086; 8883
8834 ! 3.9461574 ! 8884
8835 3 946206 618885
8836'3 9462557||8886
883713.9463048,8887
88383 9463540 1 8888
8839j3.946403l||8889
3.9484619
3.9485108
3.9485597
9486085
3.9486574
3.9487063
3.9487552
3.9488040
3.9488529
8840'3 9464523| ! 8890 3.9489018
8'34ll3.9465014
t8842 ! 3.9465505
884313.9465996
8844 ; 3.946f>437
8845|3.9466673
(8891
18892
3893
3894
8895
8846: 3.9467 40^
8347,3 9467960
884813 9468451 '
88493.9468942
88503.9469433
3.9489506
3.9489995
3.9490483
3.9490971 j
3.9491460
8961(3.9523565:
8962J3.9524049;
8963 ! 3 9524534*
8964 f 3 9525018!
8965j3. 95255Q3
8966'3.
896713.
896813
8969*3
8926 3.9506569
89273.9507055
89283 9507542
8929 3.9508028
89303.950851 5
89313.9509001
89323.9509487
8933'3.9509973
89343.9510459
8935 3.951094 6
8936 3.9511432
8937JS-9511918
39383.9512404
89393 9512889
8940'3.9513375
8941:3 9513861
8942,3.9514347
8943-3.9514832
39443.9515318 1
8945.3.9515803 1
8397
8398
8899
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3.9491948J
3.9492436
3.9492924
39493412
3.94939001
Lo^.
9518710
.9519201
.9519686
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9520656
9521141
No.
90013
9002 ! 3.
k>003'3,
!9004'3
J9005 3.
Log. f
35429081
9548390
,9543873
9544355
9544337
19006 £
.9544819
i , 521626'j9007j3 9545302
i.952211l!!9008 i 3 9546284
' 9546766
9547248
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9523080; 9010 3 <
9011 3.9547730
9012 3.9548212
9013 3.9548694
901413.9549176
9015 3 9549657
4970
8971
8972
8973
8974
8975
8976
8977
8978
8979
8981
8982
18983
8984
8985
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9526472 J9017
952695619018
9527440' ! 9019
'9527924 ;: 9020
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1.9528893 ;9022
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.9530345
3.9550139
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3.9531312
3.9531796'
3.9532280!
898013 9532763
39533247
3.9533751
3:9534214
3.9534697
3.9535181
898639535664
898713.9536147
8988'3.9536631
8989i3.9537114
8990139537597
3.9552547
'39553028
39553510
J39553991
39554472
9024
9025
902613.9554953
902713:955543 '
9028 3.9555916
9029i3 9556397
9030 ? 3.95568 78
9031(3 9557358
9032 3.9557839
9033 ! 3.9558320
90343.9558801
903539559282
8991;3'9538080
'8992J3 9538563
899313 9539046
8994!3;9539529
8995|3 9540012
8946;3.9516289j
89473.9516774
8948 3.9517260 1
8949 3.9517745]
8950I3.9518230!
8996'3 9540494
190363.9559762
9037:3.9560243
£038 39560723
19039 39561204
9040 3 9561684
899;
8998
8999
9000
3.9540977.
3-9541460
3-9541943,
3.9542425
19041 3.9562165
'904213.9562645
|9043 3 9563125
t9044!3.9563606
|9045| 3 9564 086:
19046 3.9564566
\9047\3 9565046
i 9048,3 95655i6\
, 9049 3.956600&
1905013.9566486
ARITHMETIC*.
243
«No. Log. 1
90513.9566966
9052 3.9567445
9053 39567925
90543-9568405
9055 39568885
905£'3.9569364
9057J3-9569844
9058*3.9570323
905939570803
90603.9571282
9063
906439573199
9065
9067
9068
9069
9070
906113.9571761
9062 3.9572241
3 9572720
3.9573678
9066 3.9574157
.9574636
3.9575115
3.9575594
3 9576075
9071
9072
9073
9074
9075
39576552
3.9577030
3.9577509
3.9577988
3.9578466
9076
9077
9078
9079
9080
3.9578945
3.9579423
3.9579902
3.9580380'
3.9580858
908l|3 9581337j
9082'3.9581815!
9083|39582293;
9084'3958277l!
9085J3.6583249'
908613.9583727,
9087 3 9584205
90883.9584683
9089;39585161
90903 9585639
No.,
or 3
.02*3,
03 3.
04'3,
053
063
07]3,
083,
09 ! 3
103,
No., Log. | No. | Log. j
909i;3.^8oll6
90923.9586524
9093'3.958r072 .,
909439587549 \g
9095.3 9588027;
90963.9588505
9097(3.9588982 9
90983 9589459, ]9
909913 95S9957II9
9100<3 9590414 ,| 9
9590891
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9591845
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9592800
11|3.
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13|3.
143.
153.
163^
17|3.
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20 3,
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91513.9614680
9152'3 9615160
91533 9615635
91543 9616109
91553.961658 3
91563.^617058
9157J3.9617532
9158'3.9618006
915913.9618481
9160L3 9618955
.9598043
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.9599948
9106
9167
•168
6169
9170
2i;3.
223,
23 3.
243.
25 S.
9600425
9600901
9601377
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,9602329
26|3.
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29 3,
30'3.
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9603756
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313
32i3.
33J3.
34 j 3.
353
36|a
373.
383.
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40 3.
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9605183
9605659
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41
42
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,9607561
,9608036
.9608512|
,9608987
,9609462;
96099371
9610412
96108871
,96113621
9611837]
No
J16V3.
9162*3.
9163 3
9164'S.
9165
9171
9172
9173
9174
9175
9619429
.9619903
9620387
9620851
962132*19215!
9021799
,9622272,
.9622746
.9623220!
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3.9624167
39624640
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3.9625587
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9176 3.9626534
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9201
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3-9638350
3.9638822
3.9639294
3.9639766
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3 9641653
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9265 3.9668454
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3.9648251
92233 9648722
3.964910S
3 9649664
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9230
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9232
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9267 3.9669392
9268'3.9669860
9269 3.9670326
9270 39670797
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9272.3 9671734
9273:3-9672203
9274 3.9672671
8.9660135
3.9650605
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3.9652958
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9275 3.9673139
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9277 3X674076
9278 39674544
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9280 39675480
9281
9282
9283
9284
9285
3 9076948
3.9676416
3.9676884
3.9677351
3 9677819
9286
9287
9288
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3.V078287
39678754
3.9679222
3.9679699
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3.9682027
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3.9684365
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yaui 3 yo352yo |9iUi|3.y/ U858T
5302 3.9685763 ! 9352 3 9709045
9303 3 9686230
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9305 3.9687164 t | 9355 l 3 97 10433
93Uo 39687a30 '9356 3.9? 10 JO 2,
9307 3.96880971 J9357.3 9711366
9308 39638564 ,9358 3.971 1830!
9309 3.9689030 '935^3.971 2294 !
9310 3.9689497 J9360 3.9712758!
9311 3.9689903 9361 3 97132221 9411
9312 3.9690430 |9362 3.9713686 9412
9313 3.9690896 ,9363 3.9714150 9413
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9315 3 9691829 !9365a9715078 l
9316 3.9692295, [9366 3.9715542
9317 3.9692761; 9367,3.97160051
£318 3.9693227l|9368 3.9716469
9319 3.9693694 J9369 3.9716632 (
9320 3.9694159J 9370 3.97l?396
9353;?.9709509
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No.
L<>Z
9401 3.9731741
9402
9403
9404
9405
L*4U0
9407
9408
9409
9410
39732202
3.9732664
39733126
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9321 3.9694625
9322 3 9695091
9323 3.9695557
93243.9696023
9325J3 9696488
'9371 3.9717859
19372 3 9718323
93733.9718786
J93743.9719249
937539719713
9326139696954
9327 3.9697420
9328'3.9697885
9329(3.9698351
9330J3.9698816
9331
9376,3.9720176
9377,3.9720639
93783.9721102
93793.9721565
93303.9722028
9332
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9334
9335
3.96U9282
3.9699747
3.9700213
39700678
3.9701143
93813.9722491
9382
9383
9384
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9337
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3 9702074
3 9702539
3.9703004
3 9703469
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9344
9345
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3.9703934
3.9704399
3.9704863,
3.9705328:
3.9705793;
3.9706258
3.9706722
3.9707187
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3.9736358
3.9736819
3.9737281
94143.9737742
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3.9739126
3 973958;
3.9740048
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9392 39727581
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9395 39728968
9426
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9428
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9439
9440
No,
9451
945S
9453
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3.9724050
3 9734511
3.9734973
3.9735435
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9450
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3.9755237
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3.9757534
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3.9758452
3. 9758911
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9462 3.9759829!
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3 9777693
3.9778150
3.9778607
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3.9743274
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3.9779978
3.9780435
3.9780892
3.9781348
3.9781805
3.9782263
3.9782718
3.9783175
3.9783631
3.9784088
9516
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ARITHMETICS.
245
9551
9552
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3.9805033
95623.9805487
9561
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95663.98ur304
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9570
9571
9572
9573
No.
3.9800488
3.9800945
3.9801398 f
9554 3.9801852
95553.98023071
95563.9802761J
9557a9803216'
95583.9803670
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9601
9602
9603
3.9809573
3.9810027
3.9810481
95743.98109341
95753.9811388!
9606
9607
9608
9609
9610
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3.9827686J
96123.9828138I
3.9828589
96143.9829041
9615 3.9829493
9617
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95783.9812748
9579J3 9813202
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9581|3 9814108
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9583 3.9815015
95843.9815468
958513.9815921
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9588 ; 3.9817280
958913.9817723
9590(3.98181861
9591
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3.9818639
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9593 £9819544
9594
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9597
9598
9596 3.9820902
3.9821355
8.9821807
9599 3.9822260
96003.9822712
No,
3.9823165
3.9823617!
3.9824069
96043.98245221
9605 3.9824974,
103.
3.9825426:
.9825878:
3.98263801
3.98267821
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3 9830396
3.9830848
3.9831299
3.9831751
9621
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1.9842572
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96483.9844373
96493.9844823
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3.98677171
No*| Log.
97063
97073
9708|3
970913
971013
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9872192
97563.9892718
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97163.9874875
97173.9875322
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9876663
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3.9878003
97243.9878450
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9728
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9763 3.9895833
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3.9880236
3.9880682
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97313.9881575
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3.9882467
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3.9884252
3.9384698
3.9865144
9885590
9741 3.9886035
9742.3.9886481
97433.9886927
9744I3.9887373
9745 3.9887818
9746 398S8264
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9749 3.9889601
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No. | Log.
97513.9890492
9752.'3.9890937
97533.9891382
9754 r 3. 9891828
9755(3.9892273
97663 9897167
976739897612
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3.9898501
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3.9899835
3.9900279
3.9900723
39901168
3.9901612
39902056
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3.9902944
3.9903389
J.9903833
97823.9904277
9783 3.9904721
97843.9905164
9785 3.9905608
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978759906496
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9790 3 9907827
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9793'3.9909158
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98043.9914033 98543.9936125
9805 3.9914476 .9855J39936566
9806
9807
9808
9809
3.9914919
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3.9916247
98103 9916690
9811
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9902 3.9957229'
990339957668
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3.9944051
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3 9946250
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3 9949327
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5 9886 3.9950206
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No.; JLo^.
9916 3.9909365
9917*3.9963803
99183.9964241
99193.9964679
99203.9965117
99213.9965554
9022;3.9965992:
9923 3.9966430:
3.9966867i
9925 3.9967305;
V9263.f967743
9927-3.9968180
9928;3 9968618!
9929 3.9969055!
99303.9969492
No. I Lotr.
9951 3.9978667 1
9952 3.9979104
9953 3.9979540
99543.9979*976
99543 9980413
995639980849
9957;3.9981285
995839981721
99593.9982157
6960;3|>982593
9961.39983029
9962S3.9983465
9963J3.9983901
9964i3.9984337
9965 3 9984773
9966 t o.9985209
9967J3.9985645
9968'3.9986680
9969*3.9986516
99703 9986951
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99723.9987823
3.9988258
9974*3.9988694
39989129
9971
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3.9989564
3.9990000
99783.9990435
3.9990870
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99333.9970804!; 9983 3.9992611
993439971241! 9984| 3.9993046
99S53.9971679J|
9936i3.9972ll6i
9937iS-997255a
99383.9972990
9939 3 93734271
9940 ! 3.9973864i'.
^9863.9993916
99873.9994350
9988 ! 39994785
998913.9995220
99903.9995655
9952402
9952841
P953280
9953799
9954158
.9954597
.9955036
.9955474
9955913
.9956352
99413 9974301,! 99913
99423.9974737| 9992*3
99433 9975174' 9993 3
9944-3.^75619 9994J3
9945 3 9976048 9°9r m
5J3
9996089
9996524
'9996959
9997393
9997828
9946,3 9976484 : 999613 -9998662
9947 3.9976921 j« 9997 3.9998697'
994a3 9977358ij 9998 3-9999131
9949}3.9977794t| 9999 3-9999566
9950 , 3.9978231 ''lOOOO 4.0000000
APPENDIX.
THE hyfierboliek curve was found also, by Lord Napier, the
celebrated inventor of L.Q$arithma y to afford another source for
developing and illustrating the properties and construction of
logarithms. ' For the hyperbfliick areas lying between the curve*
and one asymtote, when they are bounded by ordinates parallel
t6 the other asymtote, are analagous to the logarithms of their
abscissas or parts of the asymtote. And although Napier's
logarithms are commonly said to be the same as hyperbolick.
logarithms, it is not to be understood that hyperbolas exhibit
Napier's logarithms only, but indeed all other possible systems
of logarithms whatever. For it has been found that the loga-
rithms of the same number in two^ different systems are to one
another as the reciprocals of the logarithms of the radical num-
bers of those systems, these latter logarithms, beings taken ac-
cording to. any system whatever t
Thus*
Brigg*s logarithm of any number ;
is to JS/afiier-n logarithm of the same number: r
as JBrigg's logarithm of 1Q.\
is to Na/iier's. logarithm of 10 :
But Brigg's logarithm, or the common logarithm of 10 is 1 ,
and Napier's logarithm of 10 is 2^3Q2585693; whence if Brigg's>
or the coinmoa logarithm of any number, be denoted by c. l,
• a
" Note. This curve greatly facilitates the conception of loga*
rithms to the imagination, arid affords almost an intuitive proof,
of the very important property of -their Auctions, or very small
increments, viz. that the Auction of the number is to t&e. Auction
of the logarithm, as the number is to the subtangent... And it is
evident, that in the beginning of the generation of th«bc areas.
from the vertex of the hyperbola, the rascent increment of the
ahscisse drawn into the altitude I, is to the increment oi" the-
area,. as radius is to the angle of the ordinate and abscisse, or of.
the asymtotcs : and at the beginning of the logarithms, the ra-
ftcent increment of the natural numbers is to the increment of
tne logarithms as 1 is to the modulus of the system.
. Hence, we easily discover, that the angle formed by the asym-
totes of the hyperbola, exhibiting Brigg's System of Loga-
rithms, will be 25° 44* 25" ; this bting the angle whose bine h
Q.43429443I9, &c. the modulus of the common system.
Hi LOGAMTHMICK
and Napier's or the hyperbolick logarithm of the same nufnber>
by b. l. we shall have 2.302585095 : 1 ::*.£.: c. t. ; or
1
X. L. X X.3\Mi>B*uy5 ~ h. l. X .4342944819 = c. x. as was re*
Quired.
In comparing the different systems of logarithms, Lord Na-
pier's is evidently the mo*t simple in respect of facility of con-
ttruction, because its modulus is unity.* Thus, suppose it
were required to find the Napierean logarithm of the number 2+
Here employing the formula.
log. 2 = 2 51 -+ J-. + JL + J- + J_-f &c. I
= A-i |Bf }C f-;Df{E &c, where A is put for
f B for| A, C foi i B, D for £ C, &c.
The calculation wiil be as follows: —
A — y = .666666666 A = .6G6666666
B = $ A =* .074074074* B = .02469 13S8
C = JB' =
.008230453 fC c=
.001646091
D = *C =
.000914495 4 D =
.000130642
E=s= *D =
.000101611 £E =
.000011290
F=>£E =
.000011290 /- f F =
.000001026
G = | F =
.000001 254 tV G3BS
.000000096
H = £G =
.000000139 «VH=«
.000000009
I = fcH==
.600000015 jV 1 —
.000000001
Nap. log. 2 = .693144179
Or, retaining only eight figures* Nap. log. 2 = .69314718
Having obtained the log. of 2, we can easily find the- loga-
rithm of 4, 8, and in general of any power of 2.
Ex. 2. Required the Napierean logarithm of 5. • *
By employing the same formula as before, and proceeding;
exactly as in the last example, by taking the sum of a sufficient
* Note. Mr. Baron Maseres gives the following definition of
tfce Modulus) namely, " that it is the limit of the magnitude of a
fourth proportional to these three quantities viz. the difference
of any two natural numbers that are very nearly equal to each
other, either of the said numbers and the logarithm or measure
of the ratio they have to each othet." Or we may define the
modulus to be the natural number at that part of the system of
logarithms, where the Auction of the number is equal tothc
fluciion of the logarithm, or where the numbers and logarithms
are have equal differences. And hence it follows, that the log-
arithms of equal numbers, or of equal ratios, in different sys-
tems, arc to one another* as the moduli of those systems.
ARITHMETIC*: $4*
number of the terms of the series, we shall find the Napiereafc
log. of 5 =J= 1.609437912.
The Napierean logarithms of 2 and 5 being found* the Nap.
log. of 10 = 2X5 becomes known.
Thus, to log. 2 «' . 693J47179
Add log. 5= 1.609437912
T|tf5 sum is log. 10 = 2.302585091
Or, retaining eight figures, log, 10 == 2.30256509.
Whence also the modulus of the common system of loga-
rithms is known, for it is the reciprocal of the Napierean loga-
rithm of 10, or — ~- a .434294482
' 230258509
We can now easily find the common logarithms of the num-
bers 2 and 5 ; for yre have only to multiply the Napierean log.
already found by the modulus .434294482, or divide them by
its reciprocal 2.30258509, and the products, or quotients, arc
the logarithms sought.
Thus retaining only seven decimal places of the products, we
have,
Com. Log. 2 = .69314718 X .4342944=** .3010300
Com. Log. 5 = 1.60943791 X ,4842944 s .6989700
Com. Log. 10 =ss 2.30258509 X .4342944 = 1.0000000
Or, the Common Logarithm may be found by putting M for
.4342944, as in the following expression.
C5 3^5* 5.5* 7.6* >
And the calculation will stand th«s j
Log. 2 = 30103000
~= .17371779
5
2M
3.5*
2M ■ A
-j= .0000055$
.0023 1684
5.5*
•2M
^-j5= .00000159
2M
gJS** .00000005
Com.Log. 3= .47712126
And .47712126 *t- .4S429448 m Nap. log. 5=» 1.096SC9&.
2S0 LOGARITHMIC*
f
Another Method, to find the Logarithm of any of the natur-
al number*, 1, 2, 3, 4, Sec.
* RULE.*
1. Take the geometrical series, 1, 10, 100, 1000, 10000, 8cc.
mad apply to it the corresponding arithmetical series 1,2, 3,4,
&c. as logarithms. - . '
2. Find a geometrick mean between 1 and 10, 10 and 100>
or any other two adjaceaf terms of the series betwixt which the
proposed number lies.
S. Between the mean, thus found, and the nearest extreme,
find another geometrical mean, in the same manner ; and so on,
till you are arrived within the proposed limit of the number
whose logarithm is sought.
4. Find as many arithmetical means, in the same order as you
found the geometrical ones, and the last of these will be the
logarithm answering to the number requered.
fxamples.
Let'it be required to find the logarithm of the number 9.
Here the numbers between which 9 lies are 1 and 10.
Firit, then, the logarithm of 10 is 1, and the log. of 1 is ;
therefore _i_ =* ,5 is the arithmetical mean, and ^(1X10)
3
«^/io« 3. 1622777= geometrick mean 1 whence the loga-
rithm of 3.1622777 is .5
Secondly, the log. of 10 is 1, and the log. of 3.1622777 is 5 ;
therefore "*"- = .75 = arithmetical mean, and */ (1.0 x
3.1622777)=* 5.6234132 = geometrick mean ; whence the log*
of 5.6234132 is .75. *
Thirdly, the log. of 10 is 1, and the log. of 5.62341,32 is .75 *
1 mlm 7 1
therefore " r * — =.875 = arithmetical mean ; and */ (10 K
z
5.6234132) = 7.4989421 = geometrick mean : whence the log.
of. 7. 4989421 is .875.
Fourthly, the log of 10 is 1, and the log. of 7.4989421 is .875 ;
1 I &7<
therefore -. T '. . = .9375 = arithmetical mean, and v'OO x
2
7.4989421)= 8.6596431 = geometrick mean : whence the log*
of 8.6596431 is .9375.
* Note. The reader who wishes to inform himself more par-
ticularly concerning {he history, nature, and construction of
logarithms, may consult Hutton's Mathematical Tables, pub-
lished a few years sinc<?> where he will find his curiosity am*
ply gratified.
ARITHMETICK 2*1
fifthly, the log.of lOis 1, and the log. of 8.6596431 is .9375;
therefore 1+ * 9375 -= 96875 = arithmetical mean, and V^IO*
2
8.6596431) «= 9.3057204 = geometrick mean : whence the log.
of 9.3057204 is .96875.
Sixthly, the log. of 8.6596431 is .9375, and the logarithm of
9.3057204 is .96875 ; therefore ^ilL±-^£L 5 =.953125 «=
2
arithmetical mean, and ^(8.6596431 X 9.3057204) = 8.976*713
*■ geometrick mean : whence the log. of 8 976B713 is .953125
And, proceeding in this manner, after 25 extractions the log-
arithm of 8.9999998 will be found to be .9542425 ; which may
be taken for the logarithm of 9, hecause it differs from it onlv
by io66066 > and is therefore sufficiently exact for all practical
purposes.
And in the same manner were the logarithms of almost all
the prime numbers found by Lord Napier ; a work so incredibly
laborious, that the unremitted industry of several years
scarcely sufficient for its achievement.
j
The reader in requeued to correct the following which arc
the fifinciftal
ERRATA.
JPage 15, 26tb line from the top, for 126 gals, read 128.
16, 9 for £136,58 read £163,07
26, 27 for g 428,50 read 8 429.50
27, 28 for 182 yds. read 132 yds,
32,26 for 12114 % %\ read 1 270{4$
33, 8 for £481 read £431
40, 33 for 15* read 3*
41, 3 for 12 Ik 2 m. read 1 1 b. 58m.
ibid, t for 5 o'clock 19m. 37sec. read 6 o'clock
40m. 23, sec.
ibid: 14 for g 1940 read 8 1946
ibid. 34 for 8 50 read g 5 1 .
67, 26 for 8 975,99cta. read g 1042,86 ct&.
7*3, 26 for 73% gal. read 7£ gal.
ibid. 32 for 22 lb. read 2 lb.
74, 26 erase -2d.
ibid. 42 for ad infinetum read ad infinitum.
77,15 for Duble read Doublb.
79, 24 for fully read briefly.
84, 11 for 3, the root read 3758, the root.
87, 18 for square read cube.
95, 7 for *From the square of the height, subtract
the height, read * From the square of the distance divided by
the height, subtract the height.
110, 19 for £ 1 read£7 *
134, 9 for 64 read 40
143—147/*. for minuets read minutes.
168 11 for South-Descending read 6 Signs South-
(Descending.
191, 26 for North Ascending read North Descend-
201, 16 for right read left. \ (ing.
202, 13 erase totally.
v *
.- Af-^-
M
* ?.£ f ■
•
tv:v 8 M*
_'
f
NTV171B84
- N ' ?£Qt? t
• V
v '* , ^ ~> •> .-:
i
i .
■ i ^^;;.v. ^
•
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OCT 16 1901 • / '
•
' N ?. v 2 : *H
Arf-
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■ Mi'? i»i3 ' !
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